08 — ARCHITECTURE: the module as an engineer builds it#

NEPTUNE-II · LINUS (systems), MYCROFT's derivations cited by proposition number and never re-derived. Written 2026-09-05 at HEAD 99777ab under CHARTER.md. Pin: READ path:line @ 99777ab; docs/CEQ_SHAPE.md's own READ class is pinned to 207e7b9. Books 01–06 did not exist in docs/canon/ when this book was written (RUN ls docs/canon/ → CHARTER.md, CORRECTIONS.md), so every attachment to them cites docs/CEQ_SHAPE.md (hereafter SHAPE) and docs/PLAN.md by section and names the book that will own it.

Preface — what this book decides, and the verse that decides it first#

This book states one module, Consequence–Equilibrium Attention (CEQ), completely enough to build from the page: interfaces, tensor shapes, parameters, the forward pass, the solve, the boundary rows, the heads, the training objectives with their weights, the optimiser regime, the data interface, the inference call, the budget on the certified device and on a Kaggle card, and the determinism hooks.
The first decision is verse 08.1: the layer is a causal row-stochastic softmax head with an exact-propagation corner, and every other object in this book is a read of one lower-triangular solve on that head. If 08.1 dies, the book collapses to its Terminal — a softmax stack — and the remaining verses are withdrawn in the order of the closing table.
The second decision is 08.3: boundary conditions are identity rows of the same $P$, never a second matrix, so that $I-\gamma P$ stays a unit on the causal class by a diagonal read (SHAPE §2.2 Prop. 10).
The third is 08.4: there is one solve, $z=(I-\gamma P)^{-1}V$ with $K+2$ indicator channels appended to $V$, and every read — state, committor, displacement, safest move, phase, certificate — is a linear functional of $z$ or of one further solve.
Nothing in this book is a result. No CEQ arm has been trained (CHARTER §5 B5); every capability number is NOT MEASURED and every price at $s>64$ is a per-op floor under a $2.0\times$–$6.6\times$ dispatch gap (B14).
"Faster to train" appears only in 08.17 (Kill and Terminal) and in the closing B25 row, always as T1/T2 with the floor, $N=8$, the bed, the matched count and the depth skyline $\lfloor\log_2t^\star\rfloor+2=5$ named in the same row (CHARTER §1). "More accurate" is claimed in no verse of this book, and no A1 or A2 sentence is written here; its one occurrence is 08.6's Hypotheses, which quotes the phrase to withdraw it ("no 'more accurate' reading is licensed from it at any geometry this verse names"), and an item-6 sentence that ignored that occurrence would repeat the defect this item was rewritten to close. A1 and A2 are licensed at no geometry of this canon and this book writes neither form: 06_PREDICTIONS.md:302's Terminal reads "No 'more accurate' sentence in form A2 on any bed of this canon", and a Preface item that licensed A2 at the LM shape would contradict a neighbouring book's Terminal on the identical object. T1 and T2 are licensed at the LM shape only, and only under the counting rule 08.15's Statement now fixes and prints — a parameter that can receive no gradient in the shipped module is a buffer, applied to $g_h$ and to the two gate heads, which 08.2's blend makes provably constant in the forward at $g\equiv0$. Under that rule the LM pair reads $192$ added parameters, $0.00074627\%$, matched with $42.9\times$ of margin under Ruling 3, with no re-architecting of the control and no widening of any MLP; the widening move the previous round wrote into 08.15 is deleted, because at $8\times1025=8{,}200$ parameters it is $5.47\times$ the size of the act Ruling 3 forbids by name (READ V17K_RULINGS.md:56-59 @ 99777ab: "Do NOT re-architect to close 1.5k params"). At $d_{\rm model}=16$ the pair is unmatched under both counting rules ($0.0629\%$ and $0.7758\%$, $1.97\times$ and $24.2\times$ the bar), so T1 and T2 are unlicensed there (08.15).
Every verse ends in a Terminal within three replacements; each replacement's kill is cheaper or more decisive than the one before it.
Struck constants of STRUCK.md appear nowhere; a number without a class is a defect of this book.
Census rows closed here: B26 (the whole book, anchored at 08.1); attachment points for B13 (08.22), B14 (08.17, 08.20), B15 (08.20, 08.21), B25 (08.17, 08.20) — the same mapping the closing table prints, and the only one. The book has exactly $22$ verses, 08.1 through 08.22 (RUN grep -c '^### 08\.' docs/canon/08_ARCHITECTURE.md $\to$ 22); there is no 08.23 and no 08.24, and no verse cites one.
The reading order for a builder is 08.1 → 08.4 → 08.14 → 08.21 → 08.22; the reading order for a refuter is the closing table "Kills, cheapest first".

Part I — The layer#

08.1 — The CEQ attention layer: interface and shapes#

Statement. One layer maps $(X, \mathrm{pos}, \mathcal M, i_q, \mathcal D)$ to $(O, z, q, \Delta z, a^\dagger, \phi, \delta)$ where $X\in\mathbb R^{B\times s\times d_{\rm model}}$ are token embeddings, $\mathrm{pos}\in\{0,\dots,s-1\}$, $\mathcal M\in\{0,1\}^{B\times(K+2)\times s}$ are the set masks in the fixed channel order $(\mathrm{sink},\ \mathcal A_0,\ \mathcal A_1,\dots,\mathcal A_K)$, $i_q\in T$ is the query position, and $\mathcal D$ is the list of $m$ candidate moves. Per head the layer forms $P=W_{\beta,qk,g}$ (08.2), overwrites the boundary rows (08.3), solves once (08.4), and reads (Part II). Shapes at the design geometry and at the planned long-context geometry:

tensor	symbol	$s=64,d=16,h=1,K=2,m=8$	LM shape ($s=512$, $d_{\rm model}=512$, $h=8$, $d=64$)	S-66 long point ($s=4096$, $d=16$, $h=1$, $n=8$)
queries, keys, values	$Q,K,V$	$[B,1,64,16]$ each	$[B,8,512,64]$	$[B,1,4096,16]$
gate heads	$u,\theta$	$[B,64]$ each	$[B,512]$	$[B,4096]$
operator	$P$	$[B,1,64,64]$	$[B,8,512,512]$	$[B,1,4096,4096]$
set masks	$\mathcal M$	$[B,4,64]$	$[B,K+2,512]$	$[B,K+2,4096]$
stacked right-hand side	$\tilde V=[V\,\\|\,\mathbb 1_{\mathcal A_\bullet}]$	$[B,1,64,20]$	$[B,8,512,64+K+2]$	$[B,1,4096,16+K+2]$
state	$z$	$[B,1,64,20]$	$[B,8,512,64+K+2]$	$[B,1,4096,16+K+2]$
per-head read, pre-concatenation	$O^h$	$[B,1,64,16]$	$[B,8,512,64]$	$[B,1,4096,16]$
layer read, post-concatenation	$O$	$[B,64,16]$	$[B,512,512]$	$[B,4096,16]$
committor vector (head $0$)	$q$	$[B,4,64]$	$[B,K+2,512]$	$[B,K+2,4096]$
displacement per move (head $0$)	$\Delta z$	$[B,8,64,16]$	$[B,m,512,64]$	$[B,m,4096,16]$
move scores (head $0$)	$\max_{k\ge1}q^{(k)}(\mathrm{do}\,a)$	$[B,8]$	$[B,m]$	$[B,m]$
phase (head $0$)	$(\arg\max_k q_{i_q}^{(k)},\ \text{margin},\ \hat\tau)$	$[B]$, $[B]$, $[B]$	same	same
certificate	$\delta$ per head, route-dependent	$[h]$	$[8]$	$[1]$

Rank convention, fixed here and printed identically in 08.13 and 08.19. $O^h$ is the per-head read at $[B,h,s,d]$; $O$ is $O^h$ with the head axis concatenated, $[B,s,d_{\rm model}]$, which is what CEQAttention(...).O returns and what o_proj consumes (08.13). The API row out.O float32[B, s, d_model] (08.19) is this same object; only $O^h$ carries a head axis, and it is tabulated so the budget of 08.20 can price it.

Head convention, fixed here and carried by 08.14. $P$, $\gamma_h$, $\beta_h$, $\kappa_h$ and $\delta$ are per head; $q$, $\phi$, $\mu$, $\hat\tau$, $\Delta z$ and the move columns are not, because in a boundary layer only head $0$ carries the boundary rows and the committor head, and heads $1..h-1$ run at $\mathcal A=\{0\}$ (sink only). At $h=8$ there is therefore one committor chain per boundary layer, not eight, and no reduction over heads is defined or needed (08.14, whose Kill decides the designation). Without this rule the eight per-head $Q,R_k$ produce $h$ distinct committor vectors and 08.9's argmin, 08.10's phase and 08.19's API are unbuildable at the LM shape — which is why the rule is a Statement and not a convention.

Certificate row, in full. $\delta=0$ on route = solve_triangular, the shipped route (08.4); $\delta=\gamma^{K'+1}/(1-\gamma)$ on route = neumann_K, where $K'$ is the journalled truncation order with registered support $K'\in\{1,2,4,8,16\}$ (08.11). $K'$ (the Neumann order) and $K$ (the number of constraint sets, hence $K+2$ value channels) are two objects; CHARTER §6's scope sentence writes both with one letter, and this book does not.

The LM shape is ceq/hf/train.py::DEFAULTS (hidden_size=512, n_layers=8, n_heads=8, seq=512, batch=8, READ @ 99777ab); the S-66 points are $(s,n)\in\{(64,2048),(256,2048),(1024,128),(4096,8)\}$ (SHAPE §5.3 S-66, book 03).

Hypotheses. $\beta=1$ (regime S, SHAPE §2.1 Def. 2); $\gamma\in[0,1)$ per head; $0\in\mathcal A_{\rm sink}$; every set precedes $i_q$ and $i_q\in T$ (F1, F2); $K\ge2$ ($K=1$ makes the committor constant to $1.11\times10^{-14}$, SHAPE §5.3 S-11, RUN); $m\ge2$.

Evidence. The constructor and the three switches: READ ceq/arm_smprime.py::ArmSMPrime.__init__ @ 99777ab (wq, wk, m_head, theta_head, beta, qk, g, all nn.Parameter). The block interface: READ ceq/hf/modeling_ceq.py::CEQAttention.__init__ @ 99777ab. The dense operator at the long point is refused: $n\cdot s^2\cdot4\,\mathrm B=2048\cdot4096^2\cdot4=1.374\times10^{11}$ B $\approx137$ GB against $7.996$ GiB (DERIVED; SHAPE §2.4(c), B15), which is why $n$ is declared per $s$.

Mechanism. B26 (no architecture page); D-3 (every dial in the interface has a registered support); V-12 ($K\ge2$); V-8 (F2 as an input contract).

Kill. A forward call at the design geometry whose output tuple is missing any of the seven fields, or whose shapes disagree with the table, on the first FOUND record of the arm's kind (FOUND-not-NAMED, CHARTER §4). Instrument: the manifest field check of SHAPE §A.10 (book 04, S-01). Price: $0$ GPU-s. Planted negative: a record from ceq/arm_smprime.py::ArmSMPrime.forward, which returns [n] (one scalar at $s-1$, READ @ 99777ab) — it must fail the check, and does, because it carries no $z$, $q$, $\Delta z$ or $\phi$ (D-1, the label class the record could not leave).

If killed. The interface is reduced to the two-head form of SHAPE §2.1 Def. 7: (H-q) the committor vector by the exact solve at $\gamma=1$ on the transient block and (H-z) the state/displacement channel at $\hat\gamma$, with the move, phase and certificate fields computed outside the layer from $q$ and $z$ by the formulas of 08.9–08.11. Hypotheses: the same. Evidence: Def. 7 is the judged apparatus (SHAPE §A.2). Kill, with its dtype named (V-17): an (H-q, H-z) record whose $q$ does not satisfy the conservation row to $10^{-12}$ in float64 on the identity cell, or to $\epsilon_{32}=1.5\times10^{-4}$ in float32 on the shipped forward — decided on one draw, $0$ GPU-s, more decisive than the field check because it tests the arithmetic and not the plumbing. The float32 bar is the same DERIVED quantity 08.3 freezes and is restated here rather than carried, because a bar stated once in float64 and applied in float32 is a threshold imported out of its units: float32 unit roundoff is $2^{-24}=5.960464\times10^{-8}$ (DERIVED), so a single rounding of a value near $1$ exceeds $10^{-12}$ by $5.96\times10^{4}$ and the replacement would die on its own kill at every real draw (V-11 inverted). Without both dtypes printed the (H-q, H-z) form is not a replacement (V-9) and 08.1's chain has one link.

Terminal. A causal softmax head with a learnable $\gamma$ and one triangular solve, exposing $z$ only; the canon licenses "the layer contains softmax bitwise at $\gamma=0$" and withdraws every read-side sentence.

08.2 — The causal row-stochastic three-corner operator $P=W_{\beta,qk,g}$#

Statement. Per head, with $q_i=W_Qx_i$, $k_j=W_Kx_j$, $qk_{ij}=\kappa\,q_i\cdot k_j/\sqrt d$, gate magnitudes $m_k=\mathrm{clamp}(\mathrm{lerp}(1,u_k,g),0,1)$ and phases $\vartheta_k=g\,\theta_k$ from two per-position scalar heads $(u,\theta)=(\mathrm{Linear}_{d\to1}(x),\mathrm{Linear}_{d\to1}(x))$, $$W[i,j]=\mathbb 1[j\le i]\ \mathrm{pathProd}(m,\vartheta)_{ij}\ e^{qk_{ij}}/Z_i^{\beta},\qquad \mathrm{pathProd}(m,\vartheta)_{ij}=\prod_{k=j+1}^{i}m_ke^{\mathrm i\vartheta_k},\qquad Z_i=\sum_{j\le i}\Big(\prod_{k=j+1}^im_k\Big)e^{qk_{ij}},$$ with three scalar switches $(\beta,\kappa,g)$. The corners: $\beta=1,g=0$ is causal softmax ($m\equiv1,\vartheta\equiv0$ exactly by the blend, so $W$ is real and row-stochastic); $\beta=0,g=0$ is linear attention; $\beta=0,\kappa=0$ is the path product. The module ships in regime S at the frozen pair $(\beta=1,\ g\equiv0)$, and only there is $P$ real, causal and row-stochastic with $P_{ii}>0$ for $i\ge1$ and $P_{00}=1$; $g_h$ is non-learnable in the shipped module (08.13's Replacement 1 made the Statement of 08.13). The masking is applied before the exponential with $-\infty$ so the upper triangle is $e^{-\infty}=0$ exactly and the backward meets no inf; the division by $Z_i^\beta$ is done on real and imaginary parts separately so the real corners are bitwise. At $g\equiv0$ the operator is built and stored real — the branch if g == 0: return num.real / zb — which is what makes the $\gamma=0$ corner bitwise anyway and which fixes the byte width at $4$ B/element for 08.20 and 08.21.

Why $\beta=1$ alone does not give a row-stochastic $P$. DERIVED, three lines. (i) READ ceq/arm_smprime.py::operator @ 99777ab: num, mod = numerator(...), zb = mod.sum(-1, keepdim=True) ** beta, return torch.complex(num.real/zb, num.imag/zb); numerator returns (gh * e, rh * e) where gh = path_product(gate(m, th)) carries $e^{\mathrm i\vartheta}$ and rh = path_product(m) does not. (ii) The row sum at $\beta=1$ is therefore $\sum_{j\le i}m_{ij}e^{\mathrm i\vartheta_{ij}}e^{qk_{ij}}\big/\sum_{j\le i}m_{ij}e^{qk_{ij}}$ — the numerator carries the phase, the normaliser $Z_i$ does not — a complex number of modulus $\le1$, equal to $1$ iff $\vartheta\equiv0$, i.e. iff $g=0$. (iii) The departure is $O(1)$, not roundoff: this verse's own manifest reads the $g$ switch's effect size at $0.673101$ (Evidence below). Every verse in Parts II–IV that assumes a real row-stochastic $P$ therefore carries $g=0$ in its Hypotheses by name: 08.3 (identity rows and the conservation identity), 08.4 (the hull bound and $\Pi_\gamma$'s row sums), 08.7 ($q\in[0,1]$ and $\sum_\bullet q^{(\bullet)}=\mathbb 1$), 08.11 (the certificate's row-stochastic hypothesis), 08.12 (the absorbing-chain reading). If $g$ is ever unfrozen, none of those hypotheses hold and the operator needs a derived row-sum guard with its own frozen kill before any of them may be restated — that guard is not written in this book and is NOT MEASURED — needs the complex row-sum instrument (book 03/04).

Hypotheses. Finite logits; $\beta=1$ and $g\equiv0$ for every row-stochasticity sentence above (the two together are regime S as this book ships it); $d$ the head width; the gate is read from the key-side position $k$ (the product runs over $k\in(j,i]$); one $(u,\theta)$ pair per position shared across heads, which is the arm's own shape (READ ceq/hf/modeling_ceq.py::CEQAttention docstring @ 99777ab).

Evidence. READ ceq/arm_smprime.py:144 @ 99777ab (hop), ceq/arm_smprime.py::operator, ::numerator, ::blend @ 99777ab; LEAN lean/CEQ/V16Domain.lean:92-97 @ 99777ab (pathProd), three_corners_containment :433, corners_are_distinct :445, no_prefix_scan_represents_a_zero_gate :165 (READ via SHAPE §2.1 Def. 1). The corner is bitwise against the lane's own softmaxAttn: RUN[coord] torch.equal(O(0),PV)=True at $s=64,d=16$ (SHAPE §2.2 Prop. 1); the record's other softmax routine is $1.110223\times10^{-16}$ off on $19/64$ entries (READ V16_ARM_SMPRIME.md:266-293 @ 99777ab), so "bitwise" is always stated against a named routine. Effect sizes of the two switches in the manifest: $3.522037$ (QK), $0.673101$ ($g$) (SHAPE §A.10, RUN).

Mechanism. V-24 (the corner bind has a non-empty rejection region: $\gamma=0.5$ reads $2.3002850040264393$, $\beta=0$ at the softmax corner reads a gap $>0.5$); V-25 (the gate has a subject only in this base); P-7.

Kill. torch.equal(P_{\beta=1,g=0}, \mathrm{softmax}_{\rm causal}(qk)) reads False on any draw at $s=64,d=16$ in float32 on the certified device, against the lane's own softmaxAttn (ceq/lm.py::Attention, kind softmax). Instrument: bind B-J (SHAPE §5.3 S-20; book 04). Price: $\le1.7$ s for one cell [FITTED+RUN]. Planted negatives that must fail at $O(1)$: $\beta=0$ (gap $>0.5$), route exp_scan on a draw with a zero gate (nan, READ ceq/arm_smprime.py::hop_scan @ 99777ab).

If killed. The gate is deleted from the shipped operator: $P=\mathrm{softmax}_{\rm causal}(qk)$ with $\beta\equiv1$, $g\equiv0$ frozen (not learnable), and the only remaining parameters per head are $W_Q,W_K,W_V$ and $\gamma$. Hypotheses: finite logits. Evidence, as [V-eq], because the replacement operator is load-bearing and [V] is inadmissible for it (L-EQ, P-10) — the statement, its hypotheses, one numeric instance: statement — for a row-stochastic causal $P$ and $\gamma\in[0,1)$, the ChaCAL layer's output is $(1-\gamma)P(I-\gamma P)^{-1}V$ (fagnou-2024-chacal Eq. 5); hypotheses — $P$ causal row-stochastic, $\gamma<1$, the diagonal convention declared (ChaCAL removes it, the record's variant keeps it, SHAPE §A.4, and the two are different operators, B22); one numeric instance — at $\gamma=0$ the expression collapses to $PV$ and the lane reads torch.equal(O(0),PV)=True at $s=64,d=16$ (RUN[coord], SHAPE §2.2 Prop. 1), while at $\gamma=0.6$ every row of the bare $P(I-\gamma P)^{-1}$ sums to $1/(1-\gamma)=2.5$ (RUN[J], 08.3 Evidence), which is the same identity with the $(1-\gamma)$ factor stripped. The diagonal convention is the one line this instance does not settle and it decides two controls (B22, book 01/04). Kill: torch.equal against F.scaled_dot_product_attention(is_causal=True) reads False at $\gamma=0$ — the same bind at the same price but now with no gate in the rejection region, hence strictly more decisive about the softmax half.

Terminal. The operator is standard causal softmax; the canon withdraws every gated-corner sentence (segmentation, F0 dividend, corner 3) and keeps the resolvent read of 08.4 on it.

08.3 — Boundary rows: the goal set, $K$ constraint sets and the declared sink as identity rows#

Statement. Given disjoint position sets $\mathcal A_{\rm sink}=\{0\}$, $\mathcal A_0$ (goal), $\mathcal A_1,\dots,\mathcal A_K$ (constraints), $\mathcal A=\bigsqcup\mathcal A_\bullet$, $T=[s]\setminus\mathcal A$, the layer overwrites $P_{a,\cdot}\leftarrow e_a$ for every $a\in\mathcal A$ after the softmax, by a masked row assignment P[a,:]=0; P[a,a]=1. In canonical order $P=\begin{pmatrix}Q&R\\0&I\end{pmatrix}$ with $Q=P_{TT}$, $R_k=P_{T\mathcal A_k}$. The sink's two channel groups are separate objects and this Statement separates them. The $d$ value columns of $\tilde V$ are zeroed at position $0$, so BOS contributes nothing to $O$; the $K+2$ indicator columns are the set memberships unmodified, so $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$ and the sink is a full absorbing target of the committor read. The sink is not a goal member. Conflating the two — "the sink carries value $0$ on every indicator channel" — sets $R_{\rm sink}=0$, hence $\hat q^{(\rm sink)}_T=(I-Q)^{-1}R_{\rm sink}\mathbb 1=0$ identically and $\sum_\bullet q^{(\bullet)}=1-q^{(\rm sink)}$, which on the judge's own draw is as low as $0$ because the sink share reads $[0.362,1.000]$ (RUN[SATURN], Evidence below): 08.7's conservation kill would then fire on $100$ per cent of draws.

The object this Statement overrides, named, with the record's own inconsistency printed rather than absorbed (V-22, D-4, M-2). The sentence just called a defect is not this book's invention: it is the bed's registration, verbatim in two places. READ docs/CEQ_SHAPE.md:1673 @ 207e7b9, S-10: "BED-S therefore declares $\mathcal A_{\rm sink}=\{0\}$ with value $0$ on every indicator channel — not a goal member". READ 04_BEDS_AND_INSTRUMENTS.md:27 @ HEAD: "boundary sets $\mathcal A_{\rm sink}=\{0\}$ (value $0$ on every indicator channel)", carried again at 04_BEDS_AND_INSTRUMENTS.md:186 as a live Hypothesis of that book's own If-killed link 1. So after this Statement book 08 specifies $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$ and book 04 and S-10 specify $\mathbb 1_{\mathcal A_{\rm sink}}[0]=0$ for the same bed, and the sink share $[0.362,1.000]$ that 08.3, 08.4, 08.9 and 08.10 all cite is a RUN[SATURN] reading taken on book 04's convention. The record contradicts itself and this is decidable, not a matter of preference: DERIVED, one line — under $\mathbb 1_{\mathcal A_{\rm sink}}[0]=0$ the escape row is $R_{\rm sink}=0$, so $q^{(\rm sink)}_T=(I-Q)^{-1}\cdot0=0$ on every transient position, and a sink share of $0.362$ — let alone $1.000$ — cannot be produced by any draw of any generator under that convention. S-10's own measure line reports both the convention and the share, and the two are inconsistent inside one card. This book therefore ships the carries convention ($\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$), because it is the only one of the two under which S-10's own measured number exists.

What this book cannot close, stated as the loss it is. Which convention the canon ships is a CORRECTIONS.md row and a repoint of 04_BEDS_AND_INSTRUMENTS.md's 04.2 Statement and its :186 Hypotheses; CORRECTIONS.md is the canon's only door (CHARTER §0, §7) and no book writes into another book. That row is NOT WRITTEN — needs the coordinator's CORRECTIONS.md row naming sink_indicator_convention and repointing 04.2 (CHARTER §7). Until it exists, every committor number of this book — the sink share $[0.362,1.000]$, the conservation reading $4.4\times10^{-16}$, the $\mathcal S$-discard and $\mathcal F$-emptiness censuses of 08.9, the conditioned margin of 08.10 — is read under carries and is NOT MEASURED on the bed as registered under zeroed, which is what books 04 and S-10 currently register. sink_indicator_convention ∈ {carries, zeroed} is a mandatory manifest field (08.22), and a cell that does not journal it is a refusal, so no reader can take a number of this book for a number on the bed as registered. This half of the defect is recorded OPEN in "Attacks answered" and this verse's Terminal is in force for it. $I-\gamma P$ is then lower-triangular with diagonal $1-\gamma P_{ii}\in[1-\gamma,1)$, a unit for every $\gamma<1$; at $\gamma=1$ the transient block $I-Q$ is a unit iff $0\in\mathcal A$, because $\rho(Q)=\max_{i\in T}P_{ii}$ on the causal class.

Hypotheses. $\beta=1$ and $g\equiv0$, so $P$ is real and row-stochastic and $P_{a,\cdot}\leftarrow e_a$ leaves it so (08.2: at $g\ne0$ the rows are complex of modulus $\le1$ and none of the sentences below hold). F1: row 0 of any causal softmax at $\beta=1$ is $e_0$, so position 0 is absorbing whether or not declared. F2: walks descend, so a set after $i_q$ has committor exactly $0.0$. Every set precedes $i_q$; $i_q\in T$; sets are given as masks, never as positions derived from any solve (D-2).

Evidence. SHAPE §2.1 Def. 3 and §2.2 Prop. 10 (DERIVED + RUN): $\rho(Q)=0.6926596893360386$ read both as spectral radius and as $\max_TP_{ii}$ (RUN[J]); undeclared BOS gives $\det(I-Q)=0.0$ (RUN[J]); with $\mathcal A_{\rm sink}=\{0\}$ declared, the conservation row holds to $4.4\times10^{-16}$ and the sink share reads $[0.362,1.000]$ on the judge's draw (RUN[SATURN], SHAPE §5.3 S-10); a set at position 20 read from query 12 gives $q=0.0$ exactly (RUN[SATURN]). The bare read $P(I-\gamma P)^{-1}$ has every row sum $1/(1-\gamma)$ with or without the sets ($2.5$ at $\gamma=0.6$, RUN[J], RUN[M]), so the $(1-\gamma)$ factor is a property of the class (SHAPE §2.1 Def. 4). Boundary rows change weights downstream, never the support: $\mathrm{supp}\,\Pi_\gamma=\mathrm{supp}\,P$ (RUN[J] True, SHAPE §2.2 Prop. 5(c)). Column sink and row condition are distinct objects that coexist (V15Fork.Asink, lean/CEQ/V15Fork.lean:67-70 @ 99777ab; xiao-2023-attentionsinks [V]).

Mechanism. V-25 (a theorem whose hypothesis no draw satisfies — F1/F2 are the two census lines); V-12 (a single absorbing target makes the label constant); V-8; C8 (regime N admits no absorbing row: one_not_nilpotent, lean/CEQ/Nilpotent.lean via SHAPE §2.1 Def. 2).

Kill, with one bar per dtype, because a bar stated once and applied in two dtypes decides nothing (V-17, M-2). On the bed's own generator with $\mathcal A_{\rm sink}=\{0\}$ declared, any draw with $\rho(Q)=1$ (a second undeclared absorbing row), or the conservation residual $|q^{({\rm sink})}+q^{(0)}+\sum_kq^{(k)}-1|_\infty$ above the bar of the dtype the cell ran in, on any declared draw:

cell	dtype	frozen bar	how the bar is fixed
the identity cell (08.3, 08.7)	float64	$\epsilon_{64}=10^{-8}$	the record's own reading is $4.4\times10^{-16}$ (Evidence), eight orders below the bar
the shipped forward (08.4's `solve_triangular` on CUDA)	float32	$\epsilon_{32}=1.5\times10^{-4}$	DERIVED below
the operator row sum $\\|\mathrm{rowsum}(P)-1\\|_\infty$ (08.17's per-step guard) — a different quantity, not the conservation residual	float32	$\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}$ at $s=64$	DERIVED below, in its own units; never $\epsilon_{32}$

The third row exists because the first two bars carry an amplification the third does not, and the two were the same number until this round (V-17, V-22). DERIVED, four lines. (i) $\epsilon_{32}$ is the bar on the conservation residual of the forward substitution and its whole content is the amplification factor $|T|\cdot\|(I-Q)^{-1}\|_\infty=63\times23.81=1{,}500$; the operator's row sum carries no substitution and no such factor. (ii) With $\mathrm{rowsum}(P)=1+\eta$ on $T$, $Q\mathbb 1+R\mathbb 1=(1+\eta)\mathbb 1$, so $\sum_\bullet q=(I-Q)^{-1}R\mathbb 1=\mathbb 1+(I-Q)^{-1}\eta$ and the conservation residual is $\|(I-Q)^{-1}\|_\infty|\eta|\le23.81|\eta|$. (iii) Freezing the row-sum guard at $\epsilon_{32}=1.5\times10^{-4}$ therefore admits a conservation residual of $23.81\times1.5\times10^{-4}=3.5715\times10^{-3}$, which is $23.81\times$ this table's own float32 bar and $3.57\times10^{5}$ times its float64 bar $\epsilon_{64}=10^{-8}$ — a cell would pass the row-sum guard at every one of 08.17's $150$ steps and fire this verse's and 08.7's conservation kill on the same operator. (iv) The bar in the row sum's own units is the roundoff floor of the softmax normaliser: the row sum is $s$ additions of terms of modulus $\le1$, each carrying $2^{-24}=5.960464\times10^{-8}$, so $\epsilon_{\rm rs}=s\cdot2^{-24}=3.8147\times10^{-6}$ at $s=64$ and it is re-derived at any other $s$, never carried (V-22). Consistency, printed: at $|\eta|\le\epsilon_{\rm rs}$ the induced conservation residual is $\le23.81\times3.8147\times10^{-6}=9.083\times10^{-5}=0.606\times\epsilon_{32}$, inside this table's first two rows, which is the property the guard must have and did not. The propagated alternative $\epsilon_{32}/\|(I-Q)^{-1}\|_\infty=1.5\times10^{-4}/23.81=6.300\times10^{-6}$ is $1.65\times$ looser than $\epsilon_{\rm rs}$; the tighter of the two is frozen, so both readings are satisfied.

DERIVED, the float32 bar, three lines: (i) float32 unit roundoff is $2^{-24}=5.960464\times10^{-8}$; (ii) forward substitution over $|T|$ rows amplifies it by at most $|T|\cdot\|(I-Q)^{-1}\|_\infty$, and $\|Q\|_\infty=0.958$ (08.7 Evidence, RUN[I]) gives $\|(I-Q)^{-1}\|_\infty\le1/(1-0.958)=23.81$; (iii) at $|T|=63$ ($s=64$, one declared sink) the residual is of order $c\cdot63\cdot23.81\cdot2^{-24}=c\cdot8.94\times10^{-5}$, so $\epsilon_{32}=1.5\times10^{-4}$ freezes $c=1.68$ of headroom [ASSUMED: $c$ is the one constant of this bar and is `NOT MEASURED — needs one float32 identity cell on the certified device (book 04, S-22); until it is read, $c=1$ is the derivation and $1.68$ the headroom]. The bar rescales with $|T|$ and $\|(I-Q)^{-1}\|_\infty$ and is re-derived, never carried, at any other geometry (V-22). Applying $\epsilon_{64}=10^{-8}$ to a float32 cell fires on every honest draw — one rounding already exceeds it — and decides nothing. Instrument: bind B-E2 (SHAPE §5.3 S-22; book 04). Price: $0$ GPU-s. Planted negatives: drop BOS from every set — the solve must raise ($\det(I-Q)=0.0$), never return a number (V-16); a set after the query — $q=0.0$ exactly and the draw discarded; and, for the dtype bar itself, a float32 cell run at a geometry with $|T|=1200$ must read a residual above $\epsilon_{32}$ and be repriced rather than failed, or the bar is being carried out of its units.

If killed. Replacement 1 — the column-sink token. Delete the row overwrite; prepend one token per set whose key is trained so that softmax mass flowing to it is the absorption read, with value $0$ on the $d$ value columns and its own membership set to $1$ on its indicator column (V15Fork.Asink_computes_chain, the K-E1 control ChaCAL-with-sink-token, SHAPE §5.3 S-31). The channel separation of the Statement is inherited verbatim: a replacement that zeroes the indicator inherits the defect and is not a replacement (V-9). Hypotheses: $\beta=1$; the sink tokens precede $i_q$. Evidence: Bet F's counter predicts exactly this — component (e) is a parameterisation of a column sink (SHAPE §5.5 Bet F). Kill: the committor $\varphi$-NRMSE of the sink-token arm not within $\mathrm{MDE}_8$ of the row-boundary arm on at least $6$ of $8$ seeds (the sink token is then not a replacement and the row condition stands); price $\approx52$ s at eight seeds, escalation $\approx44$ s at $N=16$ — cheaper than nothing new but more decisive than the residual check because it is paired. Replacement 2 — the separate committor head. The layer carries no boundary rows; the committor is a second module: the exact Dirichlet solve $\hat q=(I-\hat Q)^{-1}\hat R_k\mathbb 1$ on the transient block of the learned $\hat P$ with the sets applied only inside that solve (SHAPE §2.1 Def. 7 (H-q)). Hypotheses: $0\in\mathcal A$ inside the head; $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$ on the indicator column (the Statement's separation, inherited). Evidence: bed_1.committor's form (READ ceq/beds/bed_1.py:188-198 @ 99777ab). Kill, per dtype: the head's $\hat q$ differs from the Dirichlet solve on the same $\hat Q$ by more than $10^{-12}$ in float64, or by more than $\epsilon_{32}=1.5\times10^{-4}$ in float32 — $0$ GPU-s, a pure identity. A single $10^{-12}$ bar applied to the shipped float32 path is $5.96\times10^{4}$ below one rounding and can never be met (the Kill's DERIVED block).

Terminal. A causal softmax with a declared value-zero BOS sink and no reach-avoid read; the canon licenses "the layer computes $E_i[\gamma^{\tau-1}]$ toward position 0" and withdraws "safest move under $K$ constraints".

08.4 — One lower-triangular solve with $K+2$ value channels, and the read#

Statement. With $\tilde V=[V\,\|\,\mathbb 1_{\mathcal A_{\rm sink}}\,\|\,\mathbb 1_{\mathcal A_0}\,\|\,\mathbb 1_{\mathcal A_1}\,\|\cdots\|\,\mathbb 1_{\mathcal A_K}]\in\mathbb R^{s\times(d+K+2)}$, the $d$ value columns forced to $0$ at position $0$ and the $K+2$ indicator columns left unmodified (08.3: $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$, so $R_{\rm sink}=P_{T,0}\ne0$ and $q^{(\rm sink)}$ is the escape mass, not identically zero), $$z=(I-\gamma P)^{-1}\tilde V\quad\text{by forward substitution},\qquad O=(1-\gamma)\,P\,z_{[:,:d]},\qquad \Pi_\gamma=(1-\gamma)P(I-\gamma P)^{-1}.$$ Specification of the solve (pseudocode, one head, one sequence):

# Vtilde: [s, d + K + 2].  Column groups are DISTINCT objects (08.3).
Vtilde[:, :d]   = V                     # the d VALUE columns
Vtilde[0, :d]   = 0                     # BOS contributes nothing to O
Vtilde[:, d:]   = masks.T               # the K+2 INDICATOR columns, UNMODIFIED
                                        # so Vtilde[0, d + 0] == 1  (sink membership)
                                        # R_sink = P[T, 0]  is the escape row, non-zero

M = I - gamma * P                       # lower-triangular, diag in [1-gamma, 1)
z = solve_triangular(M, Vtilde, upper=False, unitriangular=False)
O = (1 - gamma) * (P @ z[:, :d])

The row $R_{\rm sink}=P_{T,0}$, printed explicitly: with $\mathcal A_{\rm sink}=\{0\}$ its entries are the causal-softmax weights from each transient position onto BOS, strictly positive on the whole of $T$ because the mask admits $j=0$ from every $i$; zeroing the sink's indicator column would set this row's contribution to $0$ and make $q^{(\rm sink)}\equiv0$, against a measured sink share of $[0.362,1.000]$ (08.3 Evidence, RUN[SATURN]).

$\Pi_\gamma$ is row-stochastic and non-negative, so $O_i\in\mathrm{hull}\{V_j:j\le i\}$ for every $\gamma$; committors live in $[0,1]$. At $\gamma=0$, $M=I$, $z=\tilde V$ bitwise and $O=PV$ is the same matmul as attention.

Hypotheses. $\gamma\in[0,1)$; $P$ causal with finite logits (any $\beta$ for the triangularity; $\beta=1$ and $g\equiv0$ for the mixture statement, the hull bound and the $\Pi_\gamma$ row sums — at $g\ne0$ the rows are complex of modulus $\le1$, $\Pi_\gamma$ is not row-stochastic and "$O_i\in\mathrm{hull}\{V_j\}$" is false, 08.2); the solve is torch.linalg.solve_triangular in float32 on CUDA, one stream, so every parity bar of this verse is stated per dtype (08.3's Kill table).

Evidence. SHAPE §2.2 Prop. 1 (RUN[coord] torch.equal(O(0),PV)=True), Prop. 5 (DERIVED + RUN: solve_triangular vs dense inverse $1.7763568394002505\times10^{-15}$; row sums of $\Pi_\gamma$ $1.000000000000000$ RUN[F]; diagonal minimum $0.4$ at $\gamma=0.6$ RUN[M]); the cost increment over the softmax head: $+s^2d/2$ MACs, depth $s$ (DERIVED), measured $2.514-1.473=1.041$ ms/step fwd+bwd at $n=2048,s=64,d=16$ float32 (RUN NEPTUNE, per-op floor, producer owed to scripts/k_cert.py). The Lean carriers lower_triangular_isUnit, diag_one_sub_smul_pos [M], resolvent_is_triangular_solve, mixing_matrix_rowStochastic [S] are pending lake build (SHAPE §2.6 rows 5, 13; L-LEAN, P-11) and are cited as targets, never as proved.

Mechanism. M-8 (the solve is priced as an increment over the head it contains, never stand-alone); V-17 (the $(1-\gamma)$ factor and row sums in their units); V-3 (the $\gamma=0$ identity is bound with a $\gamma=0.5$ rejection region).

Kill. $\max|z_{\rm solve}-z_{\rm dense}|>10^{-12}$ in float64 on one draw at $s=64$ against torch.linalg.inv, or torch.equal(O(0),PV) False in float32. Instruments: S-20 (B-J) and B-P5 (SHAPE §5.3; book 04). Price: $\le1.7$ s. Planted negative: a dense non-triangular $M$ passed with upper=False must read $O(1)$ (SHAPE §5.3 S-20).

If killed. Replacement 1 — the Neumann route with its certificate. $z_{K'}=\sum_{t\le K'}(\gamma P)^t\tilde V$ by $K'$ matmuls, shipped only with the vector certificate $\max|z-z_{K'}|\le\gamma^{K'+1}\|\tilde V\|_\infty/(1-\gamma)$ printed with $1/(1-\gamma)$ beside it (08.11). Hypotheses: $P\ge0$ row-stochastic including identity rows. Evidence: SHAPE §2.2 Prop. 4 (equality attained on the class); one hop costs more than the solve at $s=64$ ($3.001$ vs $2.514$ ms, RUN NEPTUNE). Kill: one exceedance of the printed bound on $1{,}024$ drawn cells, or $\delta\|\tilde V\|_\infty\ge\mathrm{sd}(\text{label})$ (K-I, SHAPE §5.3 S-41); price $\approx0.04$ GPU-min — cheaper than the solve bind's cell and decisive on the route. Replacement 2 — dense inverse per chunk. Partition $[s]$ into $C$-blocks; invert each diagonal block densely ($C^3/3$ MACs) and propagate off-diagonal products — the block identity resolvent_fromBlocks [M] (SHAPE §2.6 row 6). Hypotheses: $C\le256$ so the block inverse is resident. Kill: parity against the serial solve worse than $10^{-6}$ in float32 at any $C\in\{32,64,128\}$ (N-11's PASS line inverted); price NOT MEASURED — needs the chunked kernel (book 03, N-11).

Terminal. The read is $O=PV$ at $\gamma=0$, a softmax head; the canon withdraws "one solve does what $L$ hops do" and keeps the parity sentence.

08.5 — The horizon dial $\gamma$: parameterisation, initialisation and the interpolation from softmax to the committor read#

Statement. One scalar per head, $\gamma_h=\gamma_{\max}\cdot\mathrm{clamp}(\eta_h,0,1)$ with $\gamma_{\max}=1-2^{-10}$, the raw parameter $\eta_h$ initialised at exactly $0$ so that $\gamma_h=0$ bitwise at load and $\gamma_h<1$ for every $\eta_h$, and the two-sided projection $\eta_h\leftarrow\mathrm{clip}(\eta_h,0,1)$ applied after every optimiser step, which is the repair that makes the dial's kill decide the dial. torch.clamp's backward is $\mathrm{grad}\cdot\mathbb 1[\mathrm{min}\le x\le\mathrm{max}]$ at both ends, so the trap is two-sided and a one-sided projection repairs only half of it: a single step taking $\eta_h<0$ sets the gradient to exactly $0$ and freezes $\gamma_h$ at $0$, and a single step taking $\eta_h>1$ sets the gradient to exactly $0$ and pins $\gamma_h$ at $\gamma_{\max}=0.9990234375$ for the rest of training. The upper corner is not benign: at $\gamma_{\max}$ the resolvent's conditioning is $1/(1-\gamma_{\max})=1024$, so the forward substitution over $|T|=63$ rows carries $|T|\cdot1024=64{,}512$ against float32's $2^{-24}=5.960464\times10^{-8}$, a residual of order $3.845\times10^{-3}$, which is $25.6\times$ 08.3's $\epsilon_{32}=1.5\times10^{-4}$ (DERIVED) — the upper trap therefore voids 08.3's and 08.7's conservation kill on the cell it pins, while the lower trap merely freezes the dial. The previous round's $\eta_h\leftarrow\max(\eta_h,0)$ acts at neither $\eta_h>1$ nor on the counter that decides the kill, so it repaired one boundary and left the other; it is replaced here by $\mathrm{clip}$. With $\gamma_h=0$ (or $\gamma_{\max}$) an absorbing state of the optimiser, a PINNED verdict cannot distinguish "the dial buys nothing" from "the parameterisation killed the dial" (V-9, V-10). The two-sided projection restores a two-sided gradient at both boundaries at zero forward cost; $\eta_h$ is excluded from weight decay (08.17), which otherwise shrinks any recovered positive $\eta_h$ by $(1-3\times10^{-6})$ per step toward the same trap. The equivalent smooth form, $\gamma_h=\gamma_{\max}\sigma(\eta_h)$ with $\eta_h$ initialised at a finite value putting $\gamma_h$ within one float32 ulp of $0$, is rejected here for one reason printed rather than hidden: it cannot give $\gamma_h=0$ bitwise, and the bitwise load corner of 08.13 is the book's own load rule. The dial interpolates three reads of the same solve: $\gamma=0$ — next step, $O=PV$, softmax bitwise; $0<\gamma<1$ — the hitting-time transform, $O_i=E_i[\gamma^{\tau_k-1}\mathbb 1_k]$ on an indicator channel; $\gamma\uparrow1$ — absorption probabilities, read exactly by the committor head at $\gamma=1$ on the transient block (08.7), never by pushing $\gamma_h$ to $1$ (at $\gamma=1$ with an absorbing row $I-P$ is singular). The committor head does not carry $\gamma_h$; $\gamma_h$ lives on the $z/\Delta z$ channel only.

Hypotheses. $\gamma\in[0,1)$ (a signed $\gamma$ forfeits the Neumann equality); the clamp is applied in the forward and the two-sided projection $\eta_h\leftarrow\mathrm{clip}(\eta_h,0,1)$ after every step, so $\eta_h$ never leaves $[0,1]$ and the gradient is two-sided at both boundaries. The upper corner's conditioning is printed here beside 08.3's bar, because it is the geometry at which the conservation bar is exceeded: $1/(1-\gamma_{\max})=1/2^{-10}=1024$, forward substitution over $|T|=63$ rows amplifies float32's $2^{-24}$ to $63\times1024\times5.960464\times10^{-8}=3.845\times10^{-3}$, which is $25.6\times$ $\epsilon_{32}=1.5\times10^{-4}$ (DERIVED) — so a run pinned at $\eta_h>1$ does not merely lose the dial, it produces cells 08.3's and 08.7's conservation kill must refuse. The residual corner-descent hazard — that the loss has no descent direction at $\eta=0$ — is B12's and is owned by book 02; the parameterisation hazard is repaired here rather than exported, because a kill this verse cannot decide is not this verse's kill.

Evidence. SHAPE §2.2 Prop. 9 (DERIVED corollary); Def. 7's ruling that an undiscounted label and a discounted read pull $\gamma$ in opposite directions — $q$ to $1\%$ at $\tau=8$ needs $\hat\gamma\ge0.99857$, $1/(1-\hat\gamma)\approx697$ (RUN[F]); Ruling 10′ restated for $\gamma$: the null of $\Lambda=2[\mathrm{LL}(\hat\gamma)-\mathrm{LL}(\gamma\equiv0)]$ on held-out data sits on the boundary, $95\%$ point $\tfrac12\chi^2_0+\tfrac12\chi^2_1=2.7055$, MOVED at $\Lambda>\ln n_{\rm eval}=\ln4096=8.318$ (RUN[J]; the boundary-null citation is owed, SHAPE §A.11 item 6). roffo-2026-infsa [V] learns a per-head discount in a non-causal vision read; ChaCAL fixes $\gamma=0.9$ (fagnou-2024-chacal App. C, [V]).

Mechanism. V-9 (a dial whose ablation moves nothing is a repair that changes nothing); M-2 (thresholds frozen: $2.7055$, $8.318$); M-20 (the $|\hat\gamma|<0.05$ rule is deleted — one verdict per cell); D-3.

Kill, with the discriminating clause that separates the dial from its parameterisation. Bet C's counter: $\hat\gamma$ PINNED, $\Lambda\le2.7055$ on at least $6$ of $8$ seeds of the eight-seed arena on BED-S at $t^\star=8,m=8,K=2,s=64,N=8$ (SHAPE §5.3 S-61, S-62; book 04, book 06 for the counter) — admissible only on seeds where $\eta_h$ never left $[0,1]$ at either end, and the counts are printed in the same row and journalled as two manifest fields, eta_excursions_low and eta_excursions_high (08.22), because the projection covers the lower boundary and a single pooled counter is then decided entirely by excursions the Statement's own repair makes impossible — a counter blind on the side that is not repaired is decoration (V-10). A PINNED verdict read on a seed with an excursion at either end is an optimiser artefact, is reported as such, and does not count toward the $6$ of $8$; if fewer than $6$ seeds are excursion-free the cell is repriced, not decided. Price: $0$ GPU-s on top of the arena plus a $\le17.4$ s ablation. Planted negatives, three, one per boundary plus the instrument's own: the arm at fixed $\gamma=0.9$ (ChaCAL-diag's setting) must read MOVED against $\gamma\equiv0$ on the same held-out set, or the LR instrument is blind (V-16); an arm run with the projection removed and $\eta_h$ initialised at $-10^{-6}$ must read PINNED on $8$ of $8$ with eta_excursions_low $=8$ printed, or the low counter is blind; and an arm run with the projection removed and $\eta_h$ driven above $1$ (initialised at $1+10^{-6}$) must read eta_excursions_high $=8$ with $\gamma_h$ pinned at $\gamma_{\max}=0.9990234375$ on all eight seeds and a conservation residual above $\epsilon_{32}$ on those cells (the Hypotheses' $3.845\times10^{-3}$, $25.6\times$ the bar), or the high counter is blind and half the trap is untested.

If killed. Replacement 1 — the fixed-$\gamma$ regime. $\gamma_h\equiv\gamma_{\rm env}$, a bed constant declared per task (the safety discount, fisac-2019-bridging [V]), not learned; the read is the registered discounted label $E_i[\gamma_{\rm env}^{\tau-1}\mathbb 1_k]$. Hypotheses: the task supplies $\gamma_{\rm env}$. Kill: the fixed-$\gamma$ arm's $z$-channel NRMSE not below the $\gamma\equiv0$ arm's by more than $\mathrm{MDE}_8$ on $6$ of $8$ (the dial then buys nothing even when set by hand); price inside the arena pair ($\approx34$ s), cheaper than the LR ablation and paired. Replacement 2 — the annealed dial with committor supervision. $\gamma_h$ follows a fixed schedule $0\to\gamma_{\rm env}$ over the first $S_0$ steps while the committor loss (08.17) supervises $\hat P$ through the $\gamma=1$ head, then $\eta_h$ is released. Hypotheses: $S_0\le$ half the budget. Kill: $\Lambda\le2.7055$ on $6$ of $8$ after release — the same statistic, one cell, $0$ GPU-s on top of the pair.

Terminal. $\gamma$ is a bed-side dial and regime N at $\gamma=1$ (corner 3, exact without a dial) is the only equilibrium read; the canon licenses "softmax wearing a name on this bed" (SHAPE §5.5 Bet C counter) and withdraws the horizon-dial sentence.

Part II — The reads#

08.6 — The next transient state $z$ and the mixture read $O$#

Statement. $z_i=\sum_{t\ge0}\gamma^t(P^t\tilde V)_i$ is the discounted expected future value from position $i$ under the walk $P$; $O_i=((1-\gamma)Pz)_i$ is its normalised one-step-ahead read. The layer emits $z_{[:,:d]}$ as the "next transient state" channel and $O$ as the residual-stream output (08.14). The harmonic residual $r(\hat z)=\|(I-\gamma P_{\rm env})\hat z-V\|_\infty/\|V\|_\infty$ is a score printed beside $\sigma_{\min}(I-\gamma P_{\rm env})$ and $\|I-\gamma P_{\rm env}\|_\infty\le1+\gamma$, never a loss (D-2).

Hypotheses. Regime S as 08.2 ships it ($\beta=1$ and $g\equiv0$); the bed supplies $P_{\rm env}$ for the score only; the arena at $d_{\rm model}=16$ is not a matched pair under either counting rule 08.15 prints ($0.0629\%$, $1.97\times$ Ruling 3's bar, under the shipped buffer rule; $0.7758\%$, $24.2\times$, under the every-learnable-parameter rule), so the Kill below is read as a diagnostic contrast and not as an A1 or A2 capability sentence, and no "more accurate" reading is licensed from it at any geometry this verse names. No widening of the control's MLP is invoked at any geometry of this book: Ruling 3 forbids re-architecting to close a residual by name, and 08.15 closes the LM residual by a counting rule instead (READ V17K_RULINGS.md:56-59 @ 99777ab).

Evidence. SHAPE §2.2 Prop. 3 (CLASSICAL, RUN on the arm's $P$: read vs $E[\gamma^{\tau-1}]$ to $8.33\times10^{-17}$); Prop. 5(b) (hull bound); the metric ruling of SHAPE §5.3 S-16 (position-matched NRMSE vector, field cosine, magnitude ratio; pooled Wasserstein refused because the permuted oracle scores $0.0$ against NRMSE $1.421901$, READ V20_R15_THEORY_TABLE.md:221 @ 99777ab).

Mechanism. D-1 (the label is the whole vector $z^\star$, never one coordinate); V-26 (a marginal assertion standing in for a joint claim); M-1.

Kill. Bet E's counter on BED-S: $r_{\rm softmax}/r_{\rm shape}\le2$ at marginal NRMSE within $\mathrm{MDE}_8$, $N=8$ (SHAPE §5.5 Bet E; book 04 S-62; book 06). Price: inside the $\approx34$ s pair. Planted negative: the permuted oracle must not score $0$ on the metric (S-24 plant (i)).

If killed. The $z$ channel is repriced as a cost statement: "one solve, depth $s$, $+50\%$ causal MACs on the $z$ channel alone ($s^2d/2=32{,}768$ over the head's $s^2d=65{,}536$ at $(s,d)=(64,16)$; this is the $z$-channel figure of docs/CEQ_SHAPE.md:2054, N-09, and not the read layer's total, which is $2.125\times$ the head at $(s,d,K,m,h)=(64,16,2,8,1)$ and $1.517578\times$ the head at the LM shape $(512,64,2,8,8)$ — 08.17, 08.20), returns the jointly consistent vector with residual $r$" and carries no accuracy sentence. Hypotheses: none beyond 08.4.

Kill, frozen on the object the reference measures, with both model-derived candidates computed and both shown to be already crossed (V-17, M-8). The bar this verse used to freeze — "the increment at $n=2048$ above $1.68\times$ the reference $1.041$ ms" — is a ratio of one pair of objects applied to a ratio of a different pair, and the arithmetic is printed rather than avoided. DERIVED, five lines. (i) The $1.5\times$ multiplier is (head $+$ $z$ solve)/(head) $=(s^2d+s^2d/2)/s^2d$: it is a total, not an increment. (ii) The reference $1.041$ ms is the increment $2.514-1.473$ (08.20, RUN NEPTUNE); the head is $1.473$ ms. (iii) The model's prediction for the increment is $0.5\times$ head $=0.737$ ms, while the measured increment is $1.041/1.473=0.7067$ of the head, i.e. $1.4134\times$ the model's own prediction for the same quantity — the MAC model under-predicts this measurement, so no bar built from the MAC model can sit above it. (iv) The required repair's total form, $1.5\times1.12\times1.473=2.474$ ms, is $0.984\times$ the measured total $2.514$ ms: already crossed by the reference itself. (v) Its increment form, $0.5\times1.473\times1.12=0.825$ ms, is crossed by the measured $1.041$ ms by $1.26\times$: also already crossed. Both are printed here and neither is frozen, because a bar the reference measurement fails is a kill that fires at every real draw (V-11 inverted). What is frozen: the read layer's $z$-channel measured fwd+bwd at $n=2048$, $s=64$, $d=16$ above $1.12\times$ the measured reference $2.514$ ms $=\mathbf{2.816}$ ms — the reference is the measurement, the multiplier is the clock's own $\pm12\%$ non-stationarity (08.20 Hypotheses) and nothing else. Both sides of the comparison are wall-clock on the same harness, so the $2.0\times$–$6.6\times$ dispatch band (READ scale/m3_flops.py:101-121 @ 99777ab) does not enter at all: it is a model-versus-clock band, and the sentence this verse used to print — that $1.68\times$ "sits at the bottom of the band … $0.32\times$ below its lower edge" — was self-contradictory ($1.68<2.0$ is outside the band, not at its bottom) and is deleted. The frozen bar is tighter than the one it replaces: $2.816$ ms total is an increment of $1.343$ ms against the old bar's $1.68\times1.041=1.749$ ms. Price $\approx1$ GPU-min, decisive on the only remaining claim. Planted negative: the same cell run twice under the same harness must read within $\pm12\%$ of itself, or the clock and not the code is what the bar measures.

Terminal. $z$ is exposed as a diagnostic vector; the canon licenses "the residual is a diagnostic" and withdraws "joint determination in one read".

08.7 — The committor vector $q^{(\bullet)}$: the $K+2$ channels through the exact solve at $\gamma=1$ on the transient block#

Statement. For the same $P$ with boundary rows, the committor head computes, once per forward, $$\hat q^{(k)}_T=(I-Q)^{-1}R_k\mathbb 1,\quad k\in\{{\rm sink},0,1,\dots,K\},\qquad \hat q^{(k)}_a=\mathbb 1[a\in\mathcal A_k]\ \text{for }a\in\mathcal A,$$ by one forward substitution on the $|T|\times|T|$ lower-triangular block with $K+2$ right-hand sides, and emits $q\in[0,1]^{(K+2)\times s}$ with $\sum_\bullet q^{(\bullet)}=\mathbb 1$ on $T$ (the reach-avoid identity, sets exhausting the absorbing states). The head is exact ($\delta=0$) and claims no truncation certificate; it is scored in the Fisher–Rao coordinate $\varphi(p)=2\arcsin\sqrt p$ per entry.

Hypotheses. $\beta=1$ and $g\equiv0$, so $P$ is real row-stochastic and $q\in[0,1]$ and $\sum_\bullet q^{(\bullet)}=\mathbb 1$ are statements about real numbers at all (08.2: at $g\ne0$ the rows are complex of modulus $\le1$ and the conservation identity has no left-hand side); $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$ on the indicator column, so $R_{\rm sink}\ne0$ and $q^{(\rm sink)}$ is not identically zero (08.3, 08.4); $0\in\mathcal A$ (else $I-Q$ is singular); $K\ge2$; every set before $i_q$.

Evidence. SHAPE §2.2 Prop. 3, Prop. 8, Prop. 10 (RUN[I] conservation $[0.9999999999999993,1.0]$; BED-1's real sets $A=[0],B=[1],|T|=9$: resolvent read vs bed["q"] $0.0$, residual $1.04\times10^{-17}$, Kirchhoff $<10^{-10}$, READ MATHEMATICS.md:542-600 @ 99777ab); the sup-norm Neumann bound is vacuous on the committor ($\|Q\|_\infty=0.958$, bound $19.2$ at $K'=4$ vs error $2.5\times10^{-3}$, RUN[I]), so no truncated committor ships (SHAPE §A.9).

The absorption formula as [V-eq], because $\hat q^{(k)}_T=(I-Q)^{-1}R_k\mathbb 1$ is the object every read in Parts II and III is a functional of, and [V-cat] and [U] are both weaker than [V], which is itself inadmissible for a load-bearing statement (CHARTER §2 Number rule; L-EQ, P-10). Theorem (Kemeny–Snell, absorbing chains): for a finite absorbing chain in canonical form $P=\begin{pmatrix}Q&R\\0&I\end{pmatrix}$, the matrix of absorption probabilities is $B=NR$ with $N=(I-Q)^{-1}$; the $k$-th column block gives $\hat q^{(k)}_T=(I-Q)^{-1}R_k\mathbb 1$. Hypotheses: the chain is finite; every transient state reaches some absorbing state; $Q$ is substochastic with $\rho(Q)<1$, which on the causal class is the diagonal read $\rho(Q)=\max_{i\in T}P_{ii}<1$ and is exactly the BOS-declared line of 08.3 ($\rho(Q)=0.6926596893360386$ read both ways, RUN[J]; undeclared BOS gives $\det(I-Q)=0.0$ and the solve raises). One numeric instance: BED-1's real sets, $A=[0]$, $B=[1]$, $|T|=9$ — the resolvent read against bed["q"] differs by $0.0$, residual $1.04\times10^{-17}$, Kirchhoff cross-route $<10^{-10}$ (READ MATHEMATICS.md:542-600 @ 99777ab). Or, in three lines, DERIVED without any citation: absorption from $i\in T$ happens either on the first step into $\mathcal A_k$ or after one step into $T$, so $q=R_k\mathbb 1+Qq$; hence $(I-Q)q=R_k\mathbb 1$; hence $q=(I-Q)^{-1}R_k\mathbb 1$, the inverse existing by $\rho(Q)<1$. kemeny-1960-finitemarkov [V-cat] and grinstead-1997-probability Thm 11.6 [U] are the catalogue and textbook entries and are named as such, never as the load bearer.

Mechanism. FATAL-2 of refute_instrument_occvac (the committor head does not carry $\gamma$); V-12; V-3 (the conservation row is an identity of the class and is printed including when it fails, V-23).

Kill. A committor cell journalled with route = neumann_K (no certificate exists for it; the cell is refused), or the residual $|\sum_\bullet\hat q^{(\bullet)}-1|_\infty$ above 08.3's bar for the dtype the cell ran in — $\epsilon_{64}=10^{-8}$ in float64, $\epsilon_{32}=1.5\times10^{-4}$ in float32 (08.3's Kill table and its DERIVED block) — on any declared draw (SHAPE §5.3 S-40, S-22; book 04). The single $10^{-8}$ this verse used to freeze is a float64 number and the shipped forward is float32 (08.4 Hypotheses); applied there it fires on every honest draw and decides nothing (V-17). Price: $0$ GPU-s. Planted negative: declare $\mathcal A_k$ on the wrong set — $q$ moves by $O(1)$ (SHAPE §2.2 Prop. 3 read-side plant).

If killed. The discounted read as the committor surrogate: $\tilde q^{(k)}_i=O^{(k)}_i=E_i[\gamma_{\rm env}^{\tau_k-1}\mathbb 1_k]$ from the 08.4 solve at a bed constant $\gamma_{\rm env}$, printed with the delay share $1-\sum_k\tilde q^{(k)}$ (SHAPE §2.2 Prop. 3: RUN[I] read-form $[0.227,0.567]$ at $\gamma=0.6$). Hypotheses: the task registers $\gamma_{\rm env}$ and the discounted label. Kill: $\|\tilde q-q\|_\infty>(1-\gamma_{\rm env})\max_iq_i$ on one draw — the exact gap bound $q_i-E_i[\gamma^\tau\mathbb 1_k]\ge(1-\gamma)q_i$ is then violated and the surrogate is not reading the chain; $0$ GPU-s.

Terminal. No committor read; the canon licenses "the layer computes a discounted hitting-time transform on indicator channels" and withdraws every reach-avoid sentence.

08.8 — The displacement $\Delta z$ under $\mathrm{do}(a)$: the interventional re-solve and the Sherman–Morrison column route#

Statement. A candidate move $a$ is one of two objects. (i) Query-side row clamp on the operator: $P'=P+e_{v_a}u_a^\top$ with $P'_{v_a,\cdot}=e_{u_a}$, $u_a^\top\mathbb 1=0$, $u_a\in T$ (no direct absorption, R-07). Then, with $M=(I-\gamma P)^{-1}$ and $c_a=Me_{v_a}$ (one forward-substitution column, $s^2/2$ MACs), $$\Delta z(a)=\gamma\,c_a\,\frac{u_a^\top z}{1-\gamma\,u_a^\top c_a},\qquad 1-\gamma u_a^\top c_a=\frac{1-\gamma P'_{v_av_a}}{1-\gamma P_{v_av_a}}>0\ \ \forall\gamma<1,$$ so $m$ moves cost $m\,s^2/2$ MACs plus $m\,sd$ inner products against the solve's $s^2d/2$: ratio $m/d=0.5$ at $(m,d)=(8,16)$ (Bet G). (ii) Token rewrite at position $i$: $q_i,k_i,v_i$ change, hence every row $j\ge i$; $\mathrm{rank}(\Delta P)=s-i$ and the price is the suffix re-solve $(s-i)^2d/2$ MACs, which equals the full re-solve by triangularity ($z'_{<i}=z_{<i}$). In both cases the exact identity $\Delta z=(I-\gamma P')^{-1}(\Delta\tilde V+\gamma\Delta P\,z)$ holds and $\Delta z_0=0$ on every draw. The layer's moves argument names the object per move: ("clamp", v_a, u_a) or ("rewrite", i, x_i').

Hypotheses. $\gamma<1$; clamp rows at or below $i_q$ and reachable from it (else the move is a no-op, R-08); for the rank-one formula exactly one row changes (two rows need Woodbury).

Evidence. SHAPE §2.2 Prop. 7 (DERIVED + RUN: Sherman–Morrison vs re-solve $1.45\times10^{-15}$ RUN[I], $0.511918$ RUN[F]; suffix re-solve equals full re-solve $5.6\times10^{-17}$ RUN[F]; token rewrite at $i=12$, $s=32$ gives $\mathrm{rank}(\Delta P)=20$ RUN[J]; $V\equiv\mathbb 1\Rightarrow\Delta z\le10^{-15}$, Gaussian $V$ gives $0.1096$ at $s=32$, $1.127$ at $s=64$); sherman-1950-inverse-adjustment, hager-1989-updating [V]; bottou-2013-counterfactual §7.3 is the linearisation. The value-only move leaves every $q^{(k)}$ unchanged, so the $m(K+1)$-RHS route is legal for $\Delta z$ only (refute_theory_math item 17).

Mechanism. M-8 (two objects, two prices — pricing the token rewrite at the rank-one rate is M-8 literally); P-8; V-24 (the triangularity plant: a non-causal $P$ gives $\Delta z_{<i}\ne0$); D-5.

Kill. Bet G's counter: the measured column route at or above $8\times$ one solve for $m=8$ at $n=2048,s=64,d=16$, synchronize-bracketed, median of $14$ after $2$ warm-ups (SHAPE §5.3 V-17; book 03 N-02 owns the timer). Price: seconds. Planted negative: a two-row edit must break the rank-one formula at $O(1)$ ($0.0761$, $0.0868$, RUN[SATURN]/RUN[M]) and be repaired by Woodbury.

If killed. Replacement 1 — the cached mixture. $O_{\rm cached}=\hat P_{\rm base}(I-\gamma\hat P_{\rm base})^{-1}V_{\rm int}$: $\hat P$ frozen from the un-intervened context, only $\tilde V$ re-read — the operational successor representation (momennejad-2017-sr, russek-2017-predictive, mechanism [U]), zero extra solves. Hypotheses: the move changes $V$ only. Kill (K-G1, SHAPE §5.3 S-33): on $P$-changing plants the cached arm within one seed sd of the re-solve on $6$ of $8$ — then the re-solve is a per-row control wearing a name and this replacement is the shipped channel; on $V$-only plants it must match. Price $\approx35$ s, paired, more decisive than a timer. Replacement 2 — the per-row control. $\Delta z$ is read per row from the softmax head at $\gamma=0$: $\Delta O_i=(P'V')_i-(PV)_i$; no solve. Hypotheses: none. Kill: the sign column of $\Delta O$ against the oracle below softmax's own $0.807843$ (READ MATHEMATICS.md:396-406 @ 99777ab) by more than its Clopper–Pearson half-width — $0$ GPU-s on an existing cell.

Terminal. The displacement identity is algebra of the class and the interventional channel is a cost statement; the canon licenses "consequence as a label class" and withdraws "consequence as a field the layer reads".

08.9 — The safest move: the argmin over candidate moves of the maximum constraint committor, goal first#

Statement. Per move $a\in\mathcal D$ the layer evaluates the reach-avoid vector at the query, $q_{i_q}(\mathrm{do}\,a)=(q^{({\rm sink})},q^{(0)},q^{(1)},\dots,q^{(K)})_{i_q}(a)$, by the committor head on $P'(a)$ — for a clamp, one column update of the $\gamma=1$ solve per channel; for a rewrite, one suffix re-solve — and emits three columns: $$a^\star=\arg\max_aq^{(0)}(a),\qquad a^\dagger=\arg\min_{a\in\mathcal F}\max_{k\ge1}q^{(k)}(a),\quad \mathcal F=\{a: q^{(0)}(a)\ge\max_{k\ge1}q^{(k)}(a)\},\qquad a^{\ddagger}=\arg\min_{a\in\mathcal S}\frac{\max_{k\ge1}q^{(k)}(a)}{1-q^{({\rm sink})}(a)},$$ with the lexicographic rule as a fourth column and the rule-disagreement fraction printed. The goal set exists because of the degeneracy lemma: if the constraint sets exhaust the absorbing states, $\sum_{k\ge1}q^{(k)}=\mathbb 1$ and $\max_kq^{(k)}\ge1/K$ by pigeonhole, so "avoid every constraint" is unattainable; the goal-first subject condition is how the goal enters the Chebyshev column. The rule is a one-step safety filter over evaluated candidates, never a policy. Two admission sets make the three columns total functions rather than $0/0$:

The Chebyshev admission set $\mathcal S$, because the denominator $1-q^{(\rm sink)}(a)$ attains zero on the verse's own cited draw. $\mathcal S=\{a: q^{(\rm sink)}(a)\le1-\epsilon_{\rm sink}\}$ with $\epsilon_{\rm sink}=10^{-3}$ frozen [ASSUMED: the smallest decade above the float32 conservation bar $\epsilon_{32}=1.5\times10^{-4}$ of 08.3, which it clears by $6.67\times$ (DERIVED) — a denominator $1-q^{(\rm sink)}$ below that bar is indistinguishable from zero, because $\epsilon_{32}$ is the residual the forward substitution that produces $q$ can itself carry; to be re-registered from the realised sink-share sd before any cell, as $\mu_{\rm dp}$ is]. The reason this constant used to print is false by an order of magnitude and is deleted, not softened: it read "one order below $\mu_{\rm dp}=0.1$ of 08.10", and one order below $0.1$ is $10^{-2}$, not $10^{-3}$ — a factor of $10$, carried verbatim into 08.10, 08.19, 08.22 and the closing Limits, five places holding a constant whose one printed justification did not hold at the value. The alternative reason of the required repair, "$16.8\times$ float32 unit roundoff", is the construction 08.16 uses to fix $\epsilon_q=10^{-6}$ and it lands there and not here: $16.8\times2^{-24}=1.0013\times10^{-6}$, which is $\epsilon_q$, and unit roundoff is the wrong floor for a quantity produced by a $63$-row forward substitution whose floor is $\epsilon_{32}$ — so the floor named above is $\epsilon_{32}$ and the factor is $10^{-3}/1.5\times10^{-4}=6.67$. The value $10^{-3}$ is kept and the reason is replaced; $a^\ddagger$'s ratio is thereby bounded by $\max_{k\ge1}q^{(k)}/\epsilon_{\rm sink}\le10^{3}$, which is printed as the range of the Chebyshev column. A move outside $\mathcal S$ is discarded and counted, exactly as ties within $10^{-9}$ are, and the discarded fraction is a printed census line. The guard is not optional: the sink share reads $[0.362,1.000]$ on the judge's draw (08.3 Evidence, RUN[SATURN], docs/CEQ_SHAPE.md:1673) and the upper endpoint is $1.000$, so $1-q^{(\rm sink)}=0$ and $a^\ddagger$ would be $0/0$ or $\pm\infty$ with no raise, no epsilon and no third outcome (V-16). 08.10's conditioned margin $q^{(0)}/(1-q^{(\rm sink)})$ carries the same bar, and 08.19 states the raise.

The feasible set $\mathcal F$, and what happens when it is empty. Emptiness is reachable at every $K$, not a corner: with sink, goal and $K$ constraints exhausting $\mathcal A$, $q^{(0)}+q^{(\rm sink)}+\sum_{k\ge1}q^{(k)}=1$, so $\max_{k\ge1}q^{(k)}\ge(1-q^{(0)}-q^{(\rm sink)})/K$ and $\mathcal F$ requires $q^{(0)}(a)\ge(1-q^{(\rm sink)}(a))/(K+1)$ — at $K=2$, $q^{(0)}\ge0.333\,(1-q^{(\rm sink)})$; on the judge's draw at query 20 the Chebyshev minimum is $0.000$ attained by $9$ of $20$ targets of which $7$ carry sink share above $0.5$ (Evidence below, RUN[M], R-07), which is a live emptiness case. Frozen rule: when $\mathcal F=\emptyset$ the layer emits $a^\dagger=a^\star$ with infeasible=True (08.19's out.move_cols), never a sentinel and never a silent argmin over the empty set; the infeasible fraction over the 512-draw census is printed beside the rule-disagreement fraction and enters 08.10's census kill with a frozen bar of $0.5$ — a bed on which more than half the admitted draws have no goal-dominant move is not admitted, because on it $a^\dagger$ is $a^\star$ under another name and the two columns cannot disagree.

Hypotheses. $\beta=1$, $g\equiv0$, and 08.7's (the committor is real and conserved, or none of the three columns is defined); $m_{\rm eff}=m$ (no no-op moves: clamp rows at or below $i_q$ and reachable; R-08); clamp targets $u_a\in T$ (R-07); ties within $10^{-9}$ discarded and counted; $q^{(\rm sink)}(a)\le1-\epsilon_{\rm sink}$ for admission to $\mathcal S$, $\epsilon_{\rm sink}=10^{-3}$ frozen; the empty-$\mathcal F$ rule in force. On the bed's information content: every admitted draw of BED-S satisfies $I(s_0;a^\star)=0$ by plug-in — that is admission census line (9) itself (READ docs/CEQ_SHAPE.md:1686 @ the S-12 card; docs/CEQ_SHAPE.md:1100 names it the M-21/M-18 repair) — so on every admitted draw the only floor in force is the exact zero-information floor $1-1/m=0.875$ at $m=8$. A draw with $I>0$ is a leaking draw and is discarded; the tight Fano floor's hypothesis is the negation of the admission line and is satisfied on no admitted draw.

Evidence. SHAPE §2.2 Prop. 8 (DERIVED; degeneracy lemma; floor, singular: the exact zero-information floor $1-1/m=0.875$ at $m=8$, RUN[J] — the only floor this book prints for the move columns. The tight Fano value $0.7124$ is deleted from the floor set: it is admissible only where $I(s_0;a^\star)>0$, which is the negation of BED-S admission census line (9), so it is inadmissible on every admitted draw and may be printed only as a diagnostic on the unadmitted, discarded, leaking draws where $I>0$ is measured — never as this bed's floor. The weak Fano $0.6667$ remains struck as a floor); the sink-escape census on the judge's draw: at query 20 over 20 clamp targets the Chebyshev minimum is $0.000$ attained by $9$ of $20$, of which $7$ carry sink share above $0.5$ (RUN[M], SHAPE §5.3 R-07); clamps at rows 21, 25, 31 move the query's committors by $5.551\times10^{-17}$ (RUN[M], R-08); a clamp into $\mathcal A_0$ gives $q^{(0)}=1.0000$ exactly (RUN[M], R-24). vanmoffaert-2013-chebyshev, hsu-2023-safetyfilter, summers-2010-reach-avoid [V]; misra-2023-safety-constrained-mdp [V] carried in Limits (Bellman optimality can fail in multichain constrained MDPs — the policy-level object, not claimed).

Mechanism. V-12 (with the sink declared, the Chebyshev column is minimised by "fall off the prompt" unless conditioned); V-10 (the flag rule R0 decides the goal column at zero hops on any draw where a clamp target carries the goal flag); L-FLOOR; V-17 (floors at $m_{\rm eff}$).

Which column ships, and which loss trains it — with the term through which the subject condition enters named, because a penalty that carries no gradient trains nothing. out.move returns $a^\dagger$, the constrained column (08.19). $\mathcal L_a$ of 08.16 is restated there so that its score and its label are the same object ($\hat s(a)=-\hat q^{(0)}(a)$, label $a^\star$), and a second CE term $\mathcal L_{a^\dagger}$ on the constrained column carries the constrained oracle argmin as its label; without both, the ranking the gradient moves is the unconstrained risk ranking and the shipped column is trained by nothing (D-2, M-1, V-8). The feasibility half of $\mathcal L_{a^\dagger}$ enters through the hinge $\Lambda_{\mathcal F}\,\mathrm{relu}(\max_{k\ge1}\hat q^{(k)}_{i_q}(a)-\hat q^{(0)}_{i_q}(a))$ of 08.16, whose subgradient is $\Lambda_{\mathcal F}$ on exactly the infeasible set and $0$ elsewhere. The indicator form $\Lambda_{\mathcal F}(1-\mathbb 1[a\in\hat{\mathcal F}])$ this book carried for one round is not such a term: $\partial\mathbb 1[\cdot]/\partial\hat q=0$ off a measure-zero set, so it contributed exactly $0$ to $\nabla\mathcal L_{a^\dagger}$ on $100$ per cent of draws, what remained differentiable was $\max_{k\ge1}\hat q^{(k)}$ alone — the unconstrained risk ranking — and this sentence and 08.19's were false while it stood.

Kill, in two clauses at two prices, because one of them needs the trained arm and the other does not. Clause 1 (comparator, needs the arena): R-24 frozen — the closed-form flag rule R0 ("pick the move whose clamp target carries the goal flag; else the earliest constraint-free target") at or above $\mathrm{acc}_{\rm shape}-\mathrm{MDE}_8$ on at least $6$ of $8$ arena seeds (SHAPE §5.3 R-24; book 04). Price: $0$ GPU-s for R0 itself, which is oracle-free, but $\mathrm{acc}_{\rm shape}$ and $\mathrm{MDE}_8$ come from the eight-seed arena, so the clause is decided inside the $\approx98$ s arena of 08.16 and is priced there — pricing it at $0$ GPU-s short-prices it by the whole arena and mis-sorts the closing table, which is why the table now lists it beside 08.16 with its dependency printed. Clause 2 (census, genuinely $0$ GPU-s): R0 above the exact zero-information floor $0.875$ by more than its Clopper–Pearson interval on the unrestricted bed, decided on the untrained-arm 512-draw census with no arm and no arena. Price: $0$ GPU-s. Planted negative: the honest untrained arm must read $1/m$ within its interval (S-14) — if it does not, the bed leaks and the kill is unreachable (V-11).

If killed. Replacement 1 — the committor-vector head alone. The argmin columns are deleted from the layer's outputs and the task's move label is scored only through $\varphi$-NRMSE on $q(\mathrm{do}\,a)$ per move. Hypotheses: 08.7. Kill: the $0$-hop MLP within the Clopper–Pearson half-width of the shape on $\varphi$-NRMSE at $N=8$ (K-H2, S-26) — $\approx0.5$ GPU-min, paired. Replacement 2 — the goal rule only. $a^\star=\arg\max_aq^{(0)}(a)$ with no constraint columns; the constraints enter only as absorbing rows that drain mass. Hypotheses: $K\ge2$ still (the sets must exist for the goal committor to be non-constant). Kill: the class frequency of $a^\star$ outside $(0.05,0.95)$ on the 512-draw census — $0$ GPU-s, decided before any arm.

Terminal. The layer emits $q(\mathrm{do}\,a)$ per move and no decision; the canon licenses "a reach-avoid vector per candidate" and withdraws "safest move".

08.10 — The equilibrium-phase head: which absorbing set the mass flows to, the decision point, and leaving the transient class#

Statement. At the query the phase head emits $$\phi=\arg\max_{k\in\{{\rm sink},0,\dots,K\}}q^{(k)}_{i_q},\qquad \mu=q^{(\phi)}_{i_q}-\max_{k\ne\phi}q^{(k)}_{i_q},\qquad \hat\tau=\big[(I-Q)^{-1}\mathbb 1\big]_{i_q},$$ the predicted equilibrium set, its margin, and the expected number of transient steps to absorption (the fundamental-matrix row sum). A decision point is a position with $\mu<\mu_{\rm dp}$, $\mu_{\rm dp}=0.1$ frozen [ASSUMED: one decile of the committor's range, to be re-registered from the realised margin sd before any cell]; for $K=2$ with the sink share removed this is the isocommittor $q^{(0)}/(1-q^{({\rm sink})})=\tfrac12$ band, computed only on positions admitted by 08.9's bar $q^{(\rm sink)}\le1-\epsilon_{\rm sink}$, $\epsilon_{\rm sink}=10^{-3}$ frozen — the denominator attains exactly $0$ on the judge's own draw (sink share $[0.362,1.000]$, RUN[SATURN]), and a position outside the bar is discarded and counted, never divided. A position has left the transient class when $i_q\in\mathcal A$: the head then reads $\phi$ as the set containing $i_q$, $\mu=1$, $\hat\tau=0$, and the layer flags absorbed=True — no solve is read at an absorbed query.

Hypotheses. 08.7's; $\hat\tau$ needs $\rho(Q)<1$, which is the BOS-declared line. $\hat\tau$ is a read of the arm's own chain and is not comparable to any $P_{\rm env}$-computed $\tau$, and this is stated rather than left implicit because $\hat\tau$ is one of the three components 08.12 makes "next equilibrium phase" operationally: BED-S is specified with self-loops only on declared absorbing positions, so $P_{{\rm env},ii}=0$ on $T$ (READ docs/CEQ_SHAPE.md:1679, S-11 @ 207e7b9), while the arm's operator has $P_{ii}>0$ with $\max_TP_{ii}=0.6926596893360386$ (RUN[J], 08.3). Self-loops leave absorption probabilities invariant — adding $Q_{ii}=p$ and scaling the off-diagonal row by $(1-p)$ leaves $q=(I-Q)^{-1}R\mathbb 1$ unchanged — but they do not leave $\tau=(I-Q)^{-1}\mathbb 1$ invariant, which is inflated by up to $1/(1-\max_TP_{ii})=1/0.3073403106639614=3.253722$ (DERIVED; MARS's finding prints $3.2536$ for the same quotient and the fourth decimal is corrected here rather than carried). Consequence, in force: no $\tau^\star$ label is registered — 08.18's task-record table lists z_star, q_star, dz_star, a_star, y, a_dagger_star, infeasible_star and no $\tau^\star$ — no term of 08.16 supervises $\hat\tau$, and no comparison of $\hat\tau$ against a bed $\tau$ is licensed at any geometry of this book. The route that would license one is registered rather than left open: $\tau^\star$ as an 08.18 oracle field on the self-loop-removed operator $\tilde Q_{ij}=Q_{ij}/(1-Q_{ii})$ for $i\ne j$, whose $\tau$ is comparable by construction, with its own weight in 08.16 — NOT MEASURED — needs book 04's oracle extension (S-11's label tensor) and book 02's weight. "Phase" is metastable membership (prinz-2011-msm, ramsauer-2021-hopfield [V]) and no spectral statement is made about $P$ (the Cheeger sentence at MATHEMATICS.md:455 is refused for a non-reversible $P$, B9, book 01).

Evidence. $\hat\tau=N\mathbb 1$ as expected absorption time, as [V-eq], because $\hat\tau$ is the horizon read and load-bearing, and [V-cat] and [U] are inadmissible for it (L-EQ, P-10). Theorem (Kemeny–Snell): for a finite absorbing chain with $N=(I-Q)^{-1}$, the expected number of transient steps before absorption from $i$ is $[N\mathbb 1]_i$. Hypotheses: finite chain, every transient state reaching an absorbing state, $\rho(Q)<1$ — on the causal class the diagonal read $\max_{i\in T}P_{ii}=0.6926596893360386<1$ with the sink declared (RUN[J], 08.3). One numeric instance: on BED-1's $|T|=9$ block the row sum of the same $N$ that reproduces bed["q"] to $0.0$ (residual $1.04\times10^{-17}$, READ MATHEMATICS.md:542-600 @ 99777ab) is $\hat\tau$ by the same forward substitution with $\mathbb 1$ as the right-hand side, so the instance that pins $B=NR$ pins $N\mathbb 1$ on the identical factorisation. Or, in three lines, DERIVED: from $i\in T$ the walk takes one step and then, with probability $Q_{ij}$, is at $j\in T$ with the same expectation, so $\tau=\mathbb 1+Q\tau$; hence $(I-Q)\tau=\mathbb 1$; hence $\tau=(I-Q)^{-1}\mathbb 1$. kemeny-1960-finitemarkov [V-cat] and grinstead-1997-probability Thm 11.5 [U] are the catalogue and textbook entries, named as such and not as the load bearer. The $q=\tfrac12$ isocommittor guard exists in the tree at ceq/beds/bed_1.py:5-8,:382 (READ @ 99777ab); the sink share on the judge's draw $[0.362,1.000]$ (RUN[SATURN]) shows the sink channel can dominate $\phi$, which is why the conditioned form is printed beside it.

Mechanism. V-8 (a phase label constant across the admitted query region is no label); V-12; D-3 ($\hat\tau$ is the traversal dial that must vary with $t^\star$).

Kill. On the 512-draw census at $t^\star=8,K=2,m=8,s=64$: the class frequency of $\phi$ over the admitted query region outside $(0.05,0.95)$ for any set, or $\mathrm{sd}(\mu)\le0.05$, or $\hat\tau$ read on the arm's own chain alone flat across $t^\star\in\{2,8,32\}$ (a dial that does not vary, M-3.7) — stated on the arm because no bed $\tau$ is registered to compare it against (Hypotheses), and reachable because $\hat\tau=(I-Q)^{-1}\mathbb 1$ is a function of $Q$'s off-diagonal reach, which $t^\star$ moves by construction on the untrained arm as it moves the placement of the absorbing sets: an untrained arm's $\hat\tau_{i_q}$ is expected to rise monotonically with $t^\star$ because the nearest absorbing row moves further from $i_q$, and a flat reading means the causal window is not carrying $t^\star$ at all, which is exactly 08.12's placement kill read on a second quantity — or 08.9's infeasible fraction above $0.5$ (on such a bed $\mathcal F$ is empty on more than half the admitted draws, $a^\dagger$ collapses to $a^\star$ by the frozen fallback and the two move columns cannot disagree, so the bed is not admitted), or 08.9's $\mathcal S$-discard fraction above $0.5$ (the conditioned margin is then undefined on more than half the draws) — all frozen; instrument S-12's census block extended by five lines (book 04). Price: $0$ GPU-s. Planted negative: BOS declared inside a constraint set must drive $\phi$ to that set on $100\%$ of draws.

If killed. Replacement 1 — the $K=2$ isocommittor only. $\phi\in\{\mathcal A_0,\mathcal A_1\}$, $\mu=|2q^{(0)}/(1-q^{({\rm sink})})-1|$; no $\hat\tau$. Kill: $\mathrm{sd}(\mu)\le0.05$ on the admitted region — the same census line, one set fewer. Replacement 2 — the hitting-time head only. Emit $\hat\tau$ and $q^{({\rm sink})}$; no set argmax. Kill: $\hat\tau$ flat across $t^\star$ — the same $0$ GPU-s line.

Terminal. No phase read; the canon licenses "the committor vector at the query" and withdraws "next equilibrium phase".

08.11 — The Neumann certificate as an exposed output#

Statement. Every read on a Neumann route at $K'$ hops carries the printed triple $(\delta,\ \|\tilde V\|_\infty,\ 1/(1-\gamma))$ with $$\delta_{\rm bare}=\frac{\gamma^{K'+1}}{1-\gamma},\qquad\delta_\Pi=\gamma^{K'+1},\qquad \max|z-z_{K'}|\le\delta_{\rm bare}\|\tilde V\|_\infty,$$ equality of the matrix residual attained on the class; on the exact route $\delta=0$ and the triple is still printed.

The F1 mask term, derived, with the attained value printed beside it and neither confused for the other. For $\tilde P=P-E$ with $E\ge0$ and row sums $\varepsilon$, the resolvent perturbation is $$\Big\|(I-\gamma\tilde P)^{-1}-(I-\gamma P)^{-1}\Big\|_\infty\ \le\ \frac{\gamma\,\varepsilon}{\big(1-\gamma(1-\varepsilon)\big)\,(1-\gamma)}\ =\ \delta_{\rm mask},$$ DERIVED in three lines: (i) $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ with $A=I-\gamma\tilde P$, $B=I-\gamma P$, so $B-A=-\gamma E$ and the difference is $\gamma A^{-1}EB^{-1}$; (ii) $\|E\|_\infty=\varepsilon$, $\|B^{-1}\|_\infty\le1/(1-\gamma)$ since $P$ is row-stochastic, and $\|A^{-1}\|_\infty\le1/(1-\gamma\|\tilde P\|_\infty)=1/(1-\gamma(1-\varepsilon))$ since $\tilde P$ has row sums $1-\varepsilon$; (iii) multiply. At this verse's own cited setting $s=3,\gamma=0.9,\varepsilon=0.1$ this reads $0.09/(0.19\times0.1)=4.7368421052631575$ (DERIVED). The record's RUN[J] number $0.5263157894736843$ is $\varepsilon/(1-\gamma(1-\varepsilon))=0.1/0.19$ — the attained amplification on that one instance, not a bound, and it is printed as RUN beside the DERIVED bound, $9.0\times$ below it. The naive additive $\varepsilon/(1-\gamma)=1.0$ that this verse used to state is neither: it is $4.74\times$ below the bound, so an F1-masked certificate written with it under-reports and the verse's own Kill ("one exceedance of the vector bound") would fire on the mask route by construction. The committor head prints no certificate (08.7). The certificate certifies the solve, never the model: $\hat P\ne P_{\rm env}$ has no certificate and no move is "admitted" by the threshold form.

Hypotheses. $P\ge0$ real row-stochastic including identity rows, $\gamma<1$; $\hat\beta=1$ and $g\equiv0$ — at $\hat\beta\ne1$ rows sum $1.31$–$10.29$ (READ V16_ARM_SMPRIME.md:28-32 @ 99777ab) and at $g\ne0$ the rows are complex of modulus $\le1$ (08.2), and in neither case is a certificate printed (V-25 guard). $K'$, the Neumann truncation order, has registered support $K'\in\{1,2,4,8,16\}$ — the record's own read (RUN[M], Evidence below) — and is journalled per cell (08.22's route (+K, +C) field); it is a different object from the $K$ constraint sets of the rest of the book, and the two share a letter only in CHARTER §6's scope sentence.

Evidence. SHAPE §2.2 Prop. 4 (DERIVED textbook + RUN: equality to $10^{-15}$ at $K'\in\{1,2,4,8,16\}$ with absorbing rows, RUN[M]; convergent plant rows $1.5$ at $\gamma=0.6,K'=2$: $7.29$ vs $0.54$; the attained mask amplification $0.5263157894736843$ at $s=3,\gamma=0.9,\varepsilon=0.1$, RUN[J], against the DERIVED bound $4.7368421052631575$ at the same setting — the RUN is the attained value on one instance, the DERIVED is the bound, and the Statement prints both with their classes); meyer-2000-matrix, horn-2013-matrix own the tail.

Mechanism. V-3 (declared: err = bound is an identity of the class; the bind is the plant); V-17 (vector units, $1/(1-\gamma)$ printed — the record's own factor was undefined on all eight R1 seeds, READ V15_R1.md:56 @ 99777ab); L-CERT.

Kill. K-I: one exceedance of the vector bound on $1{,}024$ drawn cells — on the mask route the bound is $\delta_{\rm bare}+\delta_{\rm mask}$ with $\delta_{\rm mask}$ the DERIVED expression above, never the naive $\varepsilon/(1-\gamma)$, which under-reports by $4.74\times$ at the verse's own setting and would make this kill fire on the mask route by construction rather than on a defect — or $\delta\|\tilde V\|_\infty\ge\mathrm{sd}(\text{label})$ (uninformative), or the row-sum identity presented as evidence about $P$ (SHAPE §5.3 S-41; book 04). Price: $\approx0.04$ GPU-min at $s=64$. Planted negatives: the convergent non-stochastic plant must fail by at least $10\times$; and an F1 mask at $\varepsilon=0.1,\gamma=0.9$ certified with the naive $1.0$ must read an exceedance while the same cell certified with the DERIVED $4.7368421052631575$ must not, or the mask term is not being tested at all.

If killed. The exact route only, $\delta=0$, route = solve_triangular on every cell; the Neumann and CSR routes are refused (L-CERT). Kill: a cell journalled with route ≠ solve_triangular — a manifest check, $0$ GPU-s.

Terminal. No certificate discipline beyond exactness; the canon licenses "the solve is exact to $1.8\times10^{-15}$" and withdraws every masked or truncated route.

Part III — Transient-state traversal#

08.12 — A sequence read as a trajectory through transient states toward an absorbing equilibrium#

Statement. Positions $0,\dots,s-1$ are the states of an absorbing chain whose transition law is $P$ (rows of $P$ move to earlier positions: the walk descends in token order). The transient class is $T$; the absorbing classes are the declared sets. Reading position $i$ "as a trajectory" means: $q_i$ is the vector of absorption probabilities from $i$; $\hat\tau_i=[(I-Q)^{-1}\mathbb 1]_i$ is the expected number of transient states visited before absorption; $z_i$ is the discounted value accumulated along the way; and the sequence of $(\phi_i,\mu_i,\hat\tau_i)$ over $i\in T$ is the trajectory's phase profile. "Next equilibrium phase" operationally is $\phi_{i_q}$ with margin $\mu_{i_q}$ and horizon $\hat\tau_{i_q}$ (08.10). The bed's environment chain is a random DAG in token order with self-loops only on declared absorbing positions, so the walk of the oracle and the walk of the arm share the causal order — both descend in token order, both are lower-triangular, both absorb on the declared sets — and they do not share the class, because they differ on the diagonal by construction. BED-S is specified with self-loops only on declared absorbing positions (READ docs/CEQ_SHAPE.md:1679, S-11), so $P_{{\rm env},ii}=0$ for every $i\in T$; the arm's operator has $P_{ii}>0$ for every $i\ge1$ (08.2's Statement), measured at $\max_TP_{ii}=\rho(Q)=0.6926596893360386$ (08.3 Evidence, RUN[J]). The named residual: $\|\hat P-P_{\rm env}\|_\infty\ge2\max_TP_{ii}=1.3853193786720772$ at the record's own reading, a lower bound attained on the diagonal alone and independent of every off-diagonal entry the arm might learn. No causal $\hat P$ with a positive diagonal is $P_{\rm env}$, and no training run closes that gap; this is the same concession the Evidence already makes for bed_1 ($\rho(Q)=0.9408612510154677$ and SymmSupport), now stated for BED-S rather than asserted away. Every kill anywhere in this book that is written on $\|\hat P-P_{\rm env}\|_\infty$ over all $(i,j)$ therefore fires by construction and is not a kill (V-11); 08.16's Replacement 1 is restated on the off-diagonal for exactly this reason (D-2 separation on an undirected substrate; lowerTriangular_ne_symmSupport [M]).

The sentence does not stop at this book, and the two registered objects it voids are named (D-4, L-G2). The record writes a kill on the identical quantity in the opposite direction: READ docs/CEQ_SHAPE.md:1687-1691 @ 207e7b9, S-13 KILL (K-D2) — "$\|\hat P-P_{\rm env}\|_\infty<10^{-3}$ on at least 6 of 8 seeds". At a floor of $1.3853193786720772$ that bar is $1{,}385\times$ unreachable, so K-D2 is not merely unfired: it is unreachable on the bed as registered and can never fire, at any seed, before or after any training run (V-11, the mirror of 08.16's Replacement 1). The same card registers "$\|\hat P-P_{\rm env}\|_\infty$ per seed as a learnability reading" on the void_contrasts list, and that reading is a diagonal constant $\ge1.385$ independent of every off-diagonal entry the arm learns — it measures the arm's own $P_{ii}$ and nothing about learning. Both are routed, not repaired here: S-13 is registered in docs/CEQ_SHAPE.md and owned by book 04, and journals never move (L-G2), so the door is a CORRECTIONS.md row naming K-D2 and the void_contrasts entry and restating both on the off-diagonal sup $\max_{i\ne j}|\hat P_{ij}-P_{{\rm env},ij}|$, which is the form 08.16's Replacement 1 already uses. That row is NOT WRITTEN — needs the coordinator's CORRECTIONS.md row (CHARTER §7); no book of this canon edits another book or the record. Until it is written, 08.22 records the full-matrix entry of void_contrasts beside its floor $1.3853193786720772$ on every cell, so no reader can take it for a measurement of learning, and this half of the defect stands OPEN in "Attacks answered" with this verse's Terminal in force for it.

Hypotheses. $\beta=1$, $g\equiv0$ (the absorbing-chain reading is a statement about a real row-stochastic $P$; 08.2); F1/F2; the label per position is computed on $P_{\rm env}$, never on $\hat P$; $t^\star$ is a tolerance dial read off the hop ladder, placed by the DAG's depth inside the causal window; shared causal order, not shared class — $P_{{\rm env},ii}=0$ on $T$ and $\hat P_{ii}>0$, so every statement below is about the two walks' common order and none about their equality. The same diagonal makes the horizon component of "next equilibrium phase" incomparable to the bed's: $q$ is invariant to self-loops but $\hat\tau=(I-Q)^{-1}\mathbb 1$ is inflated relative to any $P_{\rm env}$-computed $\tau$ by up to $1/(1-\max_TP_{ii})=3.253722$ (DERIVED, 08.10's Hypotheses), no $\tau^\star$ label is registered in 08.18, no term of 08.16 supervises $\hat\tau$, and this verse's operational definition of "next equilibrium phase" therefore reads $(\phi_{i_q},\mu_{i_q})$ against a registered label and $\hat\tau_{i_q}$ against none — a diagnostic third component, printed as one.

Evidence. SHAPE §2.2 Prop. 8 (the substrate ruling), §5.3 S-11 (the generator specification), §A.0 row 1 (three refuters killed the undirected substrate); bed_1 has $\rho(Q)=0.9408612510154677$ and SymmSupport, so no causal $\hat P$ equals it (DERIVED, refute_instrument_occvac F-B).

Mechanism. D-2; V-25; D-3 ($t^\star$ must vary); B6 (the term "state" is used in the Markov sense here and the linear-algebra sense in 08.6, SHAPE §B.6).

Kill. $t^\star=8$ unplaceable inside the causal window at $s=64$ on the 512-draw census, or the argmin error flat across $t^\star\in\{2,8,32\}$ at the one-factor sweep (M-3.7) — the traversal dial is then decoration. Price: $0$ GPU-s for placement; $\approx3.5$–$4.0$ min DERIVED for the sweep. Planted negative: a set placed after $i_q$ must read $q=0.0$ exactly and be discarded.

If killed. Replacement 1 — reroute the dial: $t^\star\to2$ or raise the DAG depth, re-run the census (SHAPE §5.10 tree C, $0$ GPU-s). Kill: still unplaceable. Replacement 2 — the jittered bed_1 landscape at $K=2$ as an oracle cross-check only, never the shape lane's substrate. Kill: by construction it cannot be the arm's class, so any capability number read on it is VOID (D-2) — the kill is a registration line, $0$ GPU-s.

Terminal. The trajectory reading is a definition on the class with no bed; the canon licenses the vocabulary of SHAPE §B.2 and withdraws every traversal number.

Part IV — Integration into a transformer block#

08.13 — The block: pre-norm residual, MLP, multi-head, and the $\gamma=0$ corner that loads a pretrained softmax stack unchanged#

Statement. One block, per the two loops in the tree:

x = x + o_proj( CEQAttention( LN1(x), masks, i_q, moves ).O )     # heads concatenated, [B,s,d_model]
x = x + MLP( LN2(x) )                                              # Linear(d, 4d) -> GELU -> Linear(4d, d)

Per head $h$: $W_Q^h,W_K^h,W_V^h\in\mathbb R^{d_{\rm model}\times d}$ (from one fused qkv linear without bias), $\gamma_h$ (08.5), $\beta_h=1$ learnable per Ruling 2 (init $1$, logged per instance — the record's granularity is one $\beta$ per layer, READ ceq/hf/modeling_ceq.py::beta_column @ 99777ab; this book specifies per head and names the difference), $\kappa_h=1$, and $g_h\equiv0$ frozen, non-learnable, in the shipped module — not "$0$ at load". Three of this book's chains depend on it and it is settled here rather than left to a replacement: (i) 08.2's row-stochasticity holds only at $g=0$, and every hypothesis of 08.3, 08.4, 08.7, 08.11 and 08.12 names it; (ii) at $g=0$ the operator is built real (08.2), which is what fixes $4$ B/element for 08.20 and 08.21; (iii) once one optimiser step moves $g$ off exactly $0.0$, the cumprod enters the forward and its cumprod→cumsum backward raises under torch.use_deterministic_algorithms(True) (READ COSTS.md §1.6 @ 99777ab), against 08.22's requirement of a bitwise deciding forward — a learnable $g$ and Ruling 1's regime cannot both hold, and this book keeps Ruling 1. The two switches per head that this Statement makes learnable are therefore $\beta_h$ and $\kappa_h$, plus $\gamma_h$; the gate heads $(u,\theta)$ are present in the module and constant in the forward at $g=0$. The shared gate heads $(u,\theta)$ are per position.

What 08.17 ships, recorded here so the two verses do not disagree on one module. 08.17's Statement is this verse's Replacement 1, not this verse's Statement: $\beta_h\equiv1$ and $\kappa_h\equiv1$ are frozen non-learnable, beta = frozen is journalled and Ruling 2 is not claimed. The reason is decidable and is printed in 08.17: a learnable $\beta_h$ receives a real gradient, $\partial W/\partial\beta=-\ln Z_i\,W$ with $\ln Z_i\approx4.1589$ at $Z_i\sim64$, so one AdamW step of magnitude $\approx\mathrm{lr}=3\times10^{-4}$ moves the row-sum residual to $4.1589\times3\times10^{-4}=1.248\times10^{-3}$, which is $327\times$ 08.17's guard bar $\epsilon_{\rm rs}=3.815\times10^{-6}$ — the guard halts training at step $1$ on this Statement's own configuration, and a weight_decay: 0.0 group does not touch that channel. The counting consequence, which 08.15 now carries as its rule, stated at the level the rule lives at. 08.15's Rule B is a rule about the module as constructed, not about the optimiser regime: an object declared a buffer at construction is not counted, an object declared an nn.Parameter at construction is counted, and a training regime that later freezes a parameter is journalled (beta = frozen, weight_decay_groups) and does not change the count. $g_h$ and the two gate heads $(u,\theta)$ are buffers at construction — 08.2's blend gives $m_k\equiv1$ and $\vartheta_k\equiv0$ for every $u_k,\theta_k$ at $g\equiv0$, so the forward is a constant function of all three and declaring them parameters would declare capacity that provably cannot exist. $\beta_h$, $\kappa_h$ and $\gamma_h$ are parameters at construction, because the forward is not constant in any of them ($\beta=0$ opens a gap $>0.5$ at the softmax corner, the $g$ switch's own effect size is $0.673101$, $\gamma=0.5$ breaks the load parity at $O(1)$), so each is a real degree of freedom of the specification even where 08.17's regime pins it. The block therefore adds $3h$ per block under Rule B — $192$ at the LM shape, $3$ at the design geometry — which is what closes the LM-shape residual at $0.00074627\%$ with no re-architecting, and leaves the design geometry unmatched at $1.97\times$ the bar. The load rule: with $\gamma_h=0$, $g_h=0$, $\beta_h=1$, $\kappa_h=1$ and $\mathcal A=\{0\}$ only, the block is bitwise the pre-norm softmax block of ceq/lm.py::Block — a pretrained softmax checkpoint loads into $W_Q,W_K,W_V$, o_proj, the two LayerNorms and the MLP with no shape change, and the new parameters ($\eta_h=0$, $u$-head bias $1$ and weights $0$, $\theta$-head $0$) add nothing to the forward. Learning proceeds from that corner.

Hypotheses. Pre-norm, eps=1e-5 (READ ceq/hf/configuration_ceq.py layer_norm_eps @ 99777ab); the pretrained stack used causal softmax with the diagonal included and no attention sink token; positions are learned embeddings of length $s$ (ceq/lm.py::TinyLM.pos).

Evidence. READ ceq/lm.py::Block, ceq/hf/modeling_ceq.py::CEQBlock @ 99777ab (pre-LN residual, Linear(d,4d)→GELU→Linear(4d,d); intermediate_size $f$ in the HF config); the corner is Prop. 1's RUN[coord] torch.equal; identity_heads() sets $m\equiv1,\theta\equiv0$ through the gate (READ ceq/arm_smprime.py::ArmSMPrime.identity_heads @ 99777ab); SMPRIME_CORNER=(1.0,1.0,1.0) is the arm's own start (READ ceq/hf/configuration_ceq.py @ 99777ab) — this book freezes $g$ at $0$ so that the hop is the all-ones mask by the blend and not by the heads, which removes the cumprod from the forward permanently and not merely at load (08.22).

Mechanism. Ruling 2/2a (no sentence transfers across corners without a bind at that corner); V-24 (the load bind's rejection region is the $\gamma=0.5$ plant and the $\beta=0$ plant); B19 (the HF package's parameter counts).

Kill. After loading a softmax checkpoint into the CEQ block at the corner, torch.equal(logits_CEQ, logits_softmax) reads False on one batch at the LM shape in float32 on the certified device; or any parameter of the loaded stack changes shape. Instrument: the identity half of B-J run at block level (book 04). Price: $\le1.7$ s. Planted negative: $\gamma_h=0.5$ on one head must break equality at $O(1)$.

If killed. Replacement 1 — the corner frozen the rest of the way. $g_h\equiv0$ is already the Statement; the replacement additionally freezes $\beta_h\equiv1$ and $\kappa_h\equiv1$ non-learnable, so only $\gamma_h$ and the projections train and the operator is causal softmax by construction at every step. Hypotheses: finite logits; Ruling 2's "$\beta$ learnable, logged per instance" is then not met and the cell is journalled beta = frozen rather than claiming Ruling 2 (V-22). Evidence: this is the ChaCAL base of 08.2's If-killed, [V-eq] there. Kill: the same loaded-corner equality at $\gamma=0$ — now with two fewer learnable parameters in the rejection region, strictly more decisive on the load; $\le1.7$ s. This replacement is what 08.17's Statement ships, and it discharges both channels of the $\beta$ hazard by construction: with $\beta_h$ frozen there is neither a decayed $\beta$ nor a gradient on $\beta$ to break row-stochasticity. It is no longer a fallback for an unimplemented parameter group — a weight_decay: 0.0 group leaves the gradient channel open and halts 08.17's guard at step $1$ (08.17's DERIVED block, $327\times$ the bar), so freezing is the shipped route and the group alone is not one. Replacement 2 — post-hoc attachment. The CEQ read is a parallel branch $x\leftarrow x+\lambda\,(O_{\rm CEQ}-O_{\rm softmax})$ with $\lambda=0$ at load; the softmax block is untouched. Hypotheses: $\lambda$ learnable from $0$. Kill: this is V-24's shape (an added branch whose proof at $\lambda=0$ never mentions the mechanism), so the bind must ship the three substitutions of MISTAKES.md V-24 — $X$ = the honest read, Gaussian noise, the label — and the noise substitution must fail the label half at $O(1)$; $0$ GPU-s.

Terminal. A pretrained softmax stack with an unused exact-propagation corner (Ruling 2's own sentence); the canon licenses "loads unchanged" and withdraws "learned from the corner".

08.14 — Which layers carry boundary rows, and how $K$ sets are declared per task#

Statement. Boundary rows are applied in every layer that carries a committor head, and the default is the last layer only ($L_{\rm b}=\{L-1\}$): earlier layers run at $\mathcal A=\{0\}$ (sink only) so that the residual stream below the read layer is the pretrained stack's. The set masks $\mathcal M\in\{0,1\}^{B\times(K+2)\times s}$ are supplied by the task per sequence (08.20); $K$ is a task constant registered with support $\{2,3,4\}$; $|\mathcal A_\bullet|\in\{1,2\}$. The query index $i_q$ is a task field; at inference on a running context $i_q=s-1$ and the layer asserts $i_q\in T$. A layer that receives no masks runs with $\mathcal A=\{0\}$ and emits $q^{({\rm sink})}$ only.

Which head carries the boundary rows, fixed here because at $h>1$ nothing else makes the read-side outputs well defined. In a boundary layer, head $0$ alone receives the row overwrite of 08.3 and carries the committor head; heads $1..h-1$ run at $\mathcal A=\{0\}$ (sink only), exactly as every non-boundary layer does. Consequences, all of them: $q$, $\Delta z$, the four move columns, $\phi$, $\mu$, $\hat\tau$ and absorbed are head-$0$ objects and carry no head axis — the ranks tabulated in 08.1 and printed in 08.19 are the ranks; $P$, $\gamma_h$, $\beta_h$, $\kappa_h$ and $\delta$ keep theirs; $O$ concatenates all $h$ heads unchanged, so the residual stream is unaffected and the load rule of 08.13 still holds. No reduction over heads is defined, because there is nothing to reduce: one boundary layer emits one committor chain. The alternative — $h$ chains at $[B,h,K+2,s]$ with a designated reduction — is rejected here and priced in the If-killed, because a mean over eight committor vectors is not a committor vector (the reach-avoid identity survives averaging, but the argmin of 08.9 and the $\arg\max$ of 08.10 do not commute with it) and no verse of this book could then say which chain $a^\dagger$ came from.

Hypotheses. 08.3's F1/F2 per layer; $\beta=1$, $g\equiv0$; the same masks in every boundary layer (a set is a property of the position, not of the layer); head $0$ of each boundary layer is the read head.

Evidence. The single-read-layer choice is the arm's own shape (one block, one read; SHAPE §B.6 "depth": one parameter block and a solve of sequential depth $s$); the sink-only default is F1 (row 0 is $e_0$ regardless) so it costs nothing to declare. Registered supports: SHAPE §5.3 S-11.

Mechanism. D-3; V-25; P-4 (a layer set "to be determined" is not a chain).

Kill, in two clauses, both of which a conforming task can fail. The discard-fraction test this verse used to freeze cannot fail on a conforming task and is therefore not a kill: 08.18 makes i_q a required field with the contract "$i_q\in T$, every set $<i_q$", and this verse's own Statement asserts $i_q\in T$, so the discard fraction is $0.0$ on $100$ per cent of draws on every bed as registered, $0.5$ below the bar, before anything runs (V-10, a gate satisfied by construction; V-11). A plant is required in addition to a reachable kill, never instead of one (V-15). The two clauses that replace it, both on the 512-draw census at $t^\star=8,K=2,m=8,s=64$, both $0$ GPU-s, both decidable before any arm:

Reachability of the boundary rows from the query, restated on the quantity that carries the claim rather than on a placement statistic, because the placement statistic fires on $100$ per cent of censuses of the bed as registered. The placement form this verse froze for one round — "the fraction of admitted draws whose read layer sees at least one declared set within $t^\star$ positions of $i_q$ falls below $0.9$" — is a kill that fires at every real draw (V-11 inverted, the mirror of the V-10 defect it was written to repair), and the DERIVED distribution is printed here rather than the bar alone. This verse's Statement fixes $i_q=s-1$ at inference; S-11's registered supports are $t^\star\in\{2,8,32\}$, $K\in\{2,3,4\}$, $|\mathcal A_\bullet|\in\{1,2\}$ (READ docs/CEQ_SHAPE.md:1679 @ 207e7b9), and S-10 places the sink at position $0$ with every other set uniformly before the query. At the census geometry $s=64$, $t^\star=8$, $K=2$, $i_q=63$ the sink sits at distance $63>8$, and the goal plus two constraint sets place $p\in[3,6]$ positions uniformly in $[1,62]$ ($62$ slots) against the $8$-slot window $[55,62]$: $P(\text{at least one in window})=1-\binom{54}{p}/\binom{62}{p}$, which is $0.34416$ at $p=3$ and $0.57987$ at $p=6$ (DERIVED) — below the frozen $0.9$ by $2.6\times$ and $1.55\times$, at every admitted $|\mathcal A_\bullet|$, before any arm. The frozen clause is therefore stated on the committor the reachability claim is about: the fraction of admitted draws on which $\max_{k\ge0}q^{(k)}_{i_q}>q^{({\rm sink})}_{i_q}$ — at least one declared non-sink set outweighs the escape at the query — falls below $\mathbf{0.216}$ [ASSUMED, with its DERIVED support printed: the event requires $q^{(\rm sink)}_{i_q}<\tfrac12$, since the $K+2$ channels sum to $1$ on $T$ (08.7); the record's only reading of that quantity is the sink share $[0.362,1.000]$ on the judge's draw (RUN[SATURN], 08.3 Evidence), whose sub-half part is $[0.362,0.5)$, i.e. $(0.5-0.362)/(1.000-0.362)=0.138/0.638=0.2163009$ of the interval on a uniform reading — so $0.216$ is the largest bar the record's own interval attains and a bed below it has a sink-share distribution strictly worse than the judge's draw; to be re-registered from S-12 census line (5)'s realised sink-share distribution before any cell, as $\mu_{\rm dp}$ and $\epsilon_{\rm sink}$ are]. A conforming task can place every set legally ($<i_q$) and still place them all outside the causal reach $t^\star$ names, in which case $q_{i_q}$ is the sink share, this clause fires, and the layer choice $L_{\rm b}=\{L-1\}$ is decoration. The distribution is NOT MEASURED — needs S-12's 512-draw census (book 04) printing the sink share apart from $q^{(0)}$, which is census line (5); until it is read the bar is [ASSUMED] at the value derived above and the clause is decided against it.
The read head carries a non-constant committor. $\mathrm{sd}(q^{(k)}_{i_q})\le0.05$ across the 512 draws for any $k$ on head $0$ of the read layer (frozen, the same bar S-12 line (5) freezes for label sd). A conforming task fails this whenever the designated read head's operator is near-uniform at $i_q$, which is a property of the layer choice and the head designation, not of the mask convention — so this clause decides the head-$0$ rule the Statement fixes.

Planted negatives, one per clause, each required to move the fraction from a value the bed actually attains and not from a bar the bed never reaches: masks shifted so every set sits at distance $t^\star+1$ from $i_q$ must drive clause 1's fraction below the census's own unshifted reading, and the unshifted reading must be printed beside it (a plant that moves the fraction from $0.9$ proves nothing, because $0.9$ is $2.6\times$ above what the geometry can produce); and a read layer whose $P$ is replaced by the uniform causal row must drive clause 2's sd to $0.0$ on all $K+2$ channels. Price: $0$ GPU-s.

If killed. Replacement 1 — boundary rows in every layer, and the head axis restored. $L_{\rm b}=\{0,\dots,L-1\}$; the residual stream carries absorption in every block; every head of every boundary layer carries the rows, so $q$ becomes $[B,h,K+2,s]$, $\Delta z$ becomes $[B,h,m,s,d]$, and $\phi,\mu,\hat\tau$ and the four move columns each gain a leading $h$ — the layer then emits $h$ chains and the reduction is declared as part of this replacement, not left open: the argmin, the argmax and the margin are computed per head and the layer returns all $h$ with no reduction, the caller choosing; a mean over heads is refused because $\arg\min$ does not commute with it. Hypotheses: the caller consumes a head axis. Kill, two clauses: the loaded-corner equality (08.13) at $\gamma=0$ still holds (identity rows at $\gamma=0$ change $O$ only on the absorbing positions themselves, which is a weight change on those rows) — the kill is the same torch.equal restricted to transient rows, $\le1.7$ s, more decisive because it tests the deeper stack; and the $h$ chains' pairwise disagreement fraction on $a^\dagger$ below $0.05$ on the 512-draw census, at which point the $h$ chains are one chain and the head axis is retained memory that buys nothing ($0$ GPU-s). Replacement 2 — sets declared by a learned membership head. $\mathcal M$ is predicted from $x$ by a sigmoid head and thresholded at $0.5$; the task supplies membership labels. Kill: this hands the arm a channel derived from a solve if the labels were oracle-derived (D-2) — the leak probe of S-15 at $R^2\ge0.5$ strikes it, $0$ GPU-s.

Terminal. The sink set only, every layer; the canon licenses "a declared BOS sink" and withdraws "$K$ sets per task".

08.15 — Matched parameters (Ruling 3): the count, and what the CEQ block adds#

Statement, opening with the counting rule, because the rule and not the arithmetic decides whether the LM pair is matched. Rule B, the rule this book ships and journals: an object the shipped forward is a constant function of is a buffer at construction and is not counted; an object the forward is not constant in is an nn.Parameter at construction and is counted, whether or not a training regime later freezes its value. It is applied to every object it reaches and to no fewer. $g_h$ is a buffer (08.13's Statement) and so are the two gate heads $(u,\theta)$ — 08.2's blend gives $m_k=\mathrm{clamp}(\mathrm{lerp}(1,u_k,0),0,1)=1$ and $\vartheta_k=0\cdot\theta_k=0$ for every $u_k,\theta_k$ at $g\equiv0$, so the forward is literally a constant function of the pair and $\partial\mathcal L/\partial u=\partial\mathcal L/\partial\theta=0$ identically. The previous round applied this rule to $g_h$ and not to the gate heads, and counted $2(d_{\rm model}+1)=1{,}026$ of the $1{,}050$ it added per block at the LM shape — $97.71$ per cent of the count — for objects it had itself declared constant in the forward; one rule, applied twice, is the whole content of this Statement. $\beta_h$, $\kappa_h$ and $\gamma_h$ are counted, because the forward is not constant in any of them ($\beta=0$ opens a gap $>0.5$ at the softmax corner, $\gamma=0.5$ breaks the load parity at $O(1)$), so 08.17's freezing of $\beta_h$ and $\kappa_h$ is an optimiser-regime field journalled as beta = frozen and weight_decay_groups = frozen_beta and changes no count — a rule that dropped a pinned degree of freedom would let a module hide arbitrary capacity behind the word "frozen". Under Rule B the CEQ block adds $h$ ($\gamma_h$) $+\,2h$ ($\beta_h,\kappa_h$) $=3h$ parameters per block and nothing else. Rule L, the alternative, printed in every row beside it and never mixed with it: every learnable nn.Parameter is counted, so the two gate heads enter at $2(d_{\rm model}+1)$ per block; this is the rule under which books 04 and 06 count $4{,}769$ and under which the record's HF pair reads $25{,}736{,}232$ (READ COSTS.md §0 via CHARTER §5 B19), and it is stated to them here so no third rule is invented downstream. params_per_arm and param_residual_pct (08.22) journal the Rule-B count with the Rule-L count printed beside it, so a cell records which rule its residual was taken under.

No re-architecting closes any residual in this book, at any geometry. The move the previous round wrote into row 5 and the If-killed — "widening the control's MLP hidden width by $1$ unit per block" — adds $8\times(2\times512+1)=8\times1025=8{,}200$ parameters (DERIVED) and is deleted. Ruling 3 verbatim, READ V17K_RULINGS.md:56-59 @ 99777ab: "0.032% residual is MATCHED. Print exact counts in every table header + one COSTS line … Do NOT re-architect to close 1.5k params — that is how new constructions sneak in." $8{,}200/1{,}500=5.47\times$ the size of the act that sentence forbids, and the act is a change to the control's architecture, which is what the sentence names. 06_PREDICTIONS.md:294 reads the identical lines as Ruling 3's "operative sentence". The residual is closed instead by stating the counting rule and applying it, which changes no operator, no control and no forward. The ratio is printed at every geometry this verse licenses, because a ratio quoted at one geometry and applied at another is a threshold out of its units (V-17), and the two bars in force are printed with it: Ruling 3's $0.032\%$, which is the bar for writing "at matched parameters" in any A1/A2/T1/T2 row (docs/CEQ_SHAPE.md:1397, :2270, R-15; 06_PREDICTIONS.md:265, :162), and S-30's $0.1\%$, which is only the admission bar for journalling a control pair at all (docs/CEQ_SHAPE.md:1909-1913; 04_BEDS_AND_INSTRUMENTS.md:130 carries both).

geometry	rule	control	shape	added	residual	vs Ruling 3 ($0.032\%$)	vs S-30 ($0.1\%$)
design, $d_{\rm model}=16$, $h=1$	B (shipped)	$4{,}769$	$4{,}772$	$3h=3$	$3/4{,}769=0.062906\%$	UNMATCHED, $1.97\times$	matched
design, $d_{\rm model}=16$, $h=1$	L	$4{,}769$	$4{,}806$	$2(d_{\rm model}{+}1)+3=37$	$37/4{,}769=0.775844\%$	UNMATCHED, $24.2\times$	UNMATCHED, $7.8\times$
design, $d_{\rm model}=16$, the record's arm	L	$4{,}769$	$4{,}806$	$37$	$0.775844\%$	UNMATCHED, $24.2\times$	UNMATCHED, $7.8\times$
LM shape, $d_{\rm model}=512$, $h=8$, $L=8$	B (shipped)	$25{,}728{,}000$	$25{,}728{,}192$	$8\times3h=8\times24=192$	$192/25{,}728{,}000=0.00074627\%$	matched, $42.9\times$ inside	matched
LM shape, per-layer switch granularity	B	$25{,}728{,}000$	$25{,}728{,}024$	$8\times3=24$	$24/25{,}728{,}000=0.00009328\%$	matched, $343\times$ inside	matched
LM shape, this book's block	L	$25{,}728{,}000$	$25{,}736{,}400$	$8\times(1026{+}8{+}16)=8{,}400$	$8{,}400/25{,}728{,}000=0.032649\%$	UNMATCHED, $1.02\times$	matched
LM shape, per-layer switch granularity	L	$25{,}728{,}000$	$25{,}736{,}232$	$8\times1029=8{,}232$	$8{,}232/25{,}728{,}000=0.031996\%$	matched, inside by $1.00012\times$	matched
LM shape, the record's HF pair	L	$25{,}728{,}000$	$25{,}736{,}232$	$8{,}232$	$0.031996\%$	matched, inside by $1.00012\times$	matched

All residuals READ/DERIVED: $37$ and $4{,}806-4{,}769$ are READ ceq/hf/modeling_ceq.py::CEQAttention docstring, confirmed at ceq/hf/modeling_ceq.py:409-418 @ 99777ab; $25{,}736{,}232$ vs $25{,}728{,}000$ is READ COSTS.md §0 via CHARTER §5 B19; $4{,}769$ is the pre-head count of READ CEQ_V20_R15_CONTRACT.md:119 @ 99777ab, on 06_PREDICTIONS.md:174's own pin, and is recorded as such below; every ratio is DERIVED arithmetic on those counts.

What the table decides, in three sentences. (a) At $d_{\rm model}=16$ the pair is unmatched under both rules — $1.97\times$ Ruling 3's bar under Rule B, $24.2\times$ under Rule L — and nothing closes it: Ruling 3's bar at that geometry is $0.032\%\times4{,}769=1.53$ parameters, below the granularity of any parameter this block carries ($1$ switch $=1$ parameter is the finest object, and even that is $0.65\times$ the bar while three of them are $1.97\times$ it), and the forbidden MLP widening would cost $2d_{\rm model}+1=33$ parameters per unit, $21.6\times$ the bar (DERIVED). Therefore T1 and T2 are unlicensed at $d_{\rm model}=16$ and every capability sentence read on the $d_{\rm model}=16$ arena is struck; the arena survives as a diagnostic cell for the kills that are not capability claims (census, identity, residual, determinism, dial), and 08.6, 08.16 and 08.17 carry the restriction by name. (b) At the LM shape the counting rule closes it and no architecture moves: Rule B reads $192$ added parameters, $0.00074627\%$, $42.9\times$ inside Ruling 3's bar — this is the closed count and the geometry at which T1 and T2 may be written; under Rule L the same block at per-layer switch granularity reads the record's own HF pair, $8{,}232$, $0.031996\%$, matched but inside by only $1.00012\times$, so the choice of rule is load-bearing and is fixed in the Statement rather than left to a reader. (c) A1 and A2 are written in no verse of this book and are licensed at no geometry of this canon: 06_PREDICTIONS.md:302's Terminal reads "No 'more accurate' sentence in form A2 on any bed of this canon", and this book's Preface item 6 now says the same rather than contradicting it.

Hypotheses. The control is the lane's own softmax block at the same $(d_{\rm model},h,L,s)$; the gate heads are shared across heads (the arm's shape); $g_h$ and the two gate heads declared buffers at construction (08.13's Statement), which is Rule B's premise and rests on 08.2's blend making the forward a constant function of all three at $g\equiv0$ — were the forward found to depend on any of them (the If-killed's parity check), Rule B would not apply to it and the row would move to Rule L's, which at per-head granularity is $8{,}464$ including $g_h$, $0.032898\%$, UNMATCHED at $1.03\times$, with no legal move left to close it. This book's block adds no readout parameters at all: 08.7's committor head is one forward substitution with no parameters, 08.9's columns are argmins, 08.10's phase is an argmax and 08.11's certificate is arithmetic, so the counts above are complete as printed and the pre-head pair is the pair this verse decides.

Evidence. READ ceq/hf/modeling_ceq.py::CEQAttention.__init__ @ 99777ab; Ruling 3 (READ V17K_RULINGS.md:56-59 @ 99777ab, verbatim: "0.032% residual is MATCHED").

Mechanism. M-8; Ruling 3; D-1 (the control must be the base case the module contains).

Kill, in two clauses at two bars, the tighter one governing the capability sentence, because one bar quoted for both sentences makes a pair at $0.05\%$ matched under this book and unmatched under book 06 (M-2, Ruling 3). The two bars are different objects and this verse freezes each against the sentence it governs: Clause 1 (the capability bar, $0.032\%$, Ruling 3, R-15, book 06) — a shape/control pair whose counts, taken under the rule the cell journals, differ by more than $0.032\%$: "at matched parameters" may not be written of it, and no T1 or T2 row may be printed on it (A1 and A2 are written in no verse of this book), which is the bar 08.6, 08.16 and 08.17 are restricted by and the bar the LM-shape row is measured against ($192$ parameters, $0.00074627\%$, $42.9\times$ inside under Rule B). The phrase "after the vector head is attached" is deleted from both clauses: no verse of this book specifies a parameterised readout, so the phrase named an object the book does not build (P-7, vocabulary with no referent), and the counts every ratio in the table is arithmetic on are pre-head counts — $4{,}769$ is the pre-head count of READ CEQ_V20_R15_CONTRACT.md:119 @ 99777ab, and READ 06_PREDICTIONS.md:174 @ HEAD records that "the post-head count with the $[m,K+2]$ readout attached is NOT MEASURED — needs book 04's recount", with 06_PREDICTIONS.md:294 adding that "the recount is the instruction, not the result, and no verse of this book may print it as performed — P-3, Ruling 3". This verse prints no post-head ratio and states that its block attaches no readout, so the pre-head pair is the pair the clause decides; a reader may not take $0.062906\%$ or $0.00074627\%$ for a post-head residual, and a post-head residual remains NOT MEASURED — needs book 04's recount. Clause 2 (the admission bar, $0.1\%$, S-30) — a pair differing by more than $0.1\%$ may not be journalled as a control pair at all, capability sentence or not; it is refused before the cell runs. The bar this verse used to freeze alone was Clause 2's, $3.1\times$ looser than Ruling 3's, quoted in the same Evidence that reads Ruling 3 verbatim — so a pair at $0.05\%$ passed this book's kill and failed 06_PREDICTIONS.md:265's requirement on the identical object (docs/CEQ_SHAPE.md:2270 R-15 freezes $0.032\%$; :1909-1913 S-30 freezes $0.1\%$; 04_BEDS_AND_INSTRUMENTS.md:130 carries both). Price: $0$ GPU-s, both clauses arithmetic. Planted negatives, one per clause: attaching the $[m,K+2]$ head to the shape only must push the difference above $0.1\%$ at $d_{\rm model}=16$ and be caught by Clause 2; and a pair constructed at exactly $0.05\%$ must pass Clause 2 and fail Clause 1, or the two bars have been collapsed into one again.

If killed — a replacement on a different object, decided by a different instrument, at a different number. The widening replacement this verse carried for one round is deleted, not retightened: its closing move is the act Ruling 3 forbids by name, at $5.47\times$ the size the ruling names, so the chain had one link and its "replacement" was the strike. The replacement derived here does not touch the control at all. Delete the two gate heads $(u,\theta)$ and $g_h$ from the module, so the shape's parameter file no longer contains them and Rule B and Rule L coincide on it: the added count is $3h$ per block under either rule — $192$ at the LM shape, $3$ at the design geometry — and no counting convention can move the pair. This is legal where widening is not: Ruling 3 forbids re-architecting to close a residual by adding parameters to a control; here the shape sheds parameters that 08.2's blend proves constant in the forward at $g\equiv0$ ($m_k\equiv1$, $\vartheta_k\equiv0$ for every $u_k,\theta_k$), so the forward is unchanged bit for bit and no new construction enters. Hypotheses: $g\equiv0$ shipped (08.13), which is what makes the deletion forward-preserving; every gated-corner sentence of the canon is already withdrawn on this route (08.2's Terminal). Evidence: 08.2's blend, READ ceq/arm_smprime.py::blend @ 99777ab; identity_heads() sets $m\equiv1,\theta\equiv0$ through the gate (READ ceq/arm_smprime.py::ArmSMPrime.identity_heads @ 99777ab). Kill — a different number, a different instrument, a different object from the Statement's: not a percentage on two integers read from a parameter file, but a bitwise forward parity on one draw — torch.equal(O_with_heads, O_without_heads) reads False at the design geometry in float32 on the certified device with $g_h=0$, against the lane's own softmaxAttn bind B-J (SHAPE §5.3 S-20; book 04). If it reads False, the heads were not constant in the forward, Rule B's premise was false for them at the Statement as well, and both the Statement's Rule-B row and this replacement fall together to the Terminal. Price: $0$ GPU-s, one draw. Planted negative: the same comparison run at $g_h=10^{-3}$ must read False at $O(1)$ (08.2's manifest effect size for the $g$ switch is $0.673101$), or the parity instrument is blind and cannot decide constancy. Strictly more decisive than the Statement's clauses because it tests the forward the parameters are counted for, and not the file they are counted in.

Terminal. Counts printed, "matched" withdrawn; every A1/A2 sentence is then unlicensed on the pair.

Part V — Training#

08.16 — The training objectives and their weights#

Statement. On a task supplying $(z^\star,\ q^\star,\ \Delta z^\star(a),\ a^\star,\ a^{\dagger\star},\ \text{violation flags } y_k(a))$ the loss on one sequence is $$\mathcal L=\lambda_z\mathcal L_z+\lambda_q\mathcal L_q+\lambda_\Delta\mathcal L_\Delta+\lambda_c\mathcal L_c+\lambda_a\mathcal L_a+\lambda_{a^\dagger}\mathcal L_{a^\dagger}+\lambda_{\rm LM}\mathcal L_{\rm LM},$$ $$\mathcal L_z=\frac{1}{|T_{\ge i_{\min}}|\,d}\sum_{i\ge i_{\min}}\frac{\|\hat z_i-z^\star_i\|_2^2}{\sigma_z^2},\qquad \mathcal L_q=\frac{1}{|T|(K+2)}\sum_{i\in T}\sum_k\big(\varphi(\hat q^{(k)}_i)-\varphi(q^{\star(k)}_i)\big)^2,$$ $$\mathcal L_\Delta=\frac{1}{m}\sum_a\Big(1-\cos\big(\Delta\hat z(a),\Delta z^\star(a)\big)\Big)+\frac{1}{m}\sum_a\frac{\|\Delta\hat z(a)-\Delta z^\star(a)\|_2^2}{\|\Delta z^\star(a)\|_2^2+\epsilon_\Delta},\qquad \epsilon_\Delta=10^{-8}\ \text{frozen},$$ $$\mathcal L_c=\frac{1}{mK}\sum_a\sum_{k\ge1}\mathrm{BCE}\big(\mathrm{clamp}(\hat q^{(k)}_{i_q}(a),\ \epsilon_q,\ 1-\epsilon_q),\,y_k(a)\big),\qquad \epsilon_q=10^{-6}\ \text{frozen},$$ $$\mathcal L_a=\mathrm{CE}\big(\mathrm{softmax}_a(-\hat s(a)/\tau_a),\,a^\star\big),\quad \hat s(a)=-\hat q^{(0)}_{i_q}(a),\qquad \mathcal L_{a^\dagger}=\mathrm{CE}\big(\mathrm{softmax}_a(-\hat s^\dagger(a)/\tau_a),\,a^{\dagger\star}\big),\quad \hat s^\dagger(a)=\max_{k\ge1}\hat q^{(k)}_{i_q}(a)+\Lambda_{\mathcal F}\,\mathrm{relu}\Big(\max_{k\ge1}\hat q^{(k)}_{i_q}(a)-\hat q^{(0)}_{i_q}(a)\Big),$$ $\mathcal L_{\rm LM}$ the byte-level next-token cross-entropy where the task has tokens to predict.

The two guards without which two of the seven terms cannot be evaluated at all, each with its value and its class, because a symbol in a loss with no number and no class is not a loss (P-1, V-16). The form this verse used to carry wrote a bare $\epsilon$ in $\mathcal L_\Delta$'s denominator — no value, no support, no evidence class, while every other constant beside it carried [ASSUMED] — and wrote $\mathcal L_c$'s BCE on the raw head output; both are defects and both are repaired here rather than left to a replacement.

$\epsilon_\Delta=10^{-8}$ frozen [ASSUMED: a denominator guard sized to be invisible on every non-degenerate label and decisive on none; the record's own displacement magnitudes are $0.1096$ at $s=32$ and $1.127$ at $s=64$ (08.8 Evidence, RUN[F]/RUN[J]), so $\|\Delta z^\star\|_2^2$ is of order $10^{-2}$ to $1$ and $10^{-8}$ shifts the denominator by $10^{-6}$ of itself — under one float32 ulp, $2^{-24}=5.960464\times10^{-8}$ relative; to be re-registered from the realised $\|\Delta z^\star\|_2^2$ distribution before any cell, as $\tau_a$ and $\Lambda_{\mathcal F}$ are]. What it does not do is make the term meaningful at the degenerate label, and the book does not pretend otherwise: $\Delta z^\star\equiv0$ exactly at $\gamma_h=0$, and for a clamp at $u_a^\top z=0$ (08.19 clause 1 as restated; not at $\Delta\tilde V\equiv0$, which holds on every clamp and predicts nothing), so on every step of stage A the denominator is $\epsilon_\Delta$ alone and the ratio reads $\|\Delta\hat z\|_2^2/10^{-8}$ — which is why the $\Delta z$ term is refused per call on 08.19's dz_degenerate flag and counted in dz_degenerate_frac (Hypotheses below), and not left to the guard to keep finite. The guard's job is the $0/0$ that survives that refusal, never the analytic zero.
$\epsilon_q=10^{-6}$ frozen, the BCE input clamp $\hat q\leftarrow\mathrm{clamp}(\hat q,\epsilon_q,1-\epsilon_q)$. Without it $\mathcal L_c$ is non-finite on the verse's own cited draws, because $\hat q$ is produced by an exact linear solve and attains both endpoints: a clamp into $\mathcal A_0$ gives $q^{(0)}=1.0000$ exactly (RUN[M], R-24, 08.9 Evidence) and at query 20 the Chebyshev minimum is $0.000$ attained by $9$ of $20$ targets (RUN[M], R-07), so $\log(0)=-\infty$ enters the sum on those draws. DERIVED, the value: the clamp caps a single term at $-\log\epsilon_q=13.8155$, and $\epsilon_q=10^{-6}$ sits $16.8\times$ above float32 unit roundoff $2^{-24}=5.960464\times10^{-8}$, so an interior committor cannot be pushed onto the clamp by rounding, while an exact endpoint always is. The alternative — a logit head scored by binary_cross_entropy_with_logits — is rejected and the reason printed: the committor head is an exact solve returning probabilities in $[0,1]$ (08.7), not logits, and inserting a logit parameterisation between the solve and the loss would break the Dirichlet consistency the next paragraph asserts. Both constants are mandatory manifest fields (epsilon_dz, epsilon_q, 08.22), so a cell run at another value is a different cell and is journalled as one.

Score and label are the same object in each move term, and the column the API ships is trained by one of them. The form this verse used to carry — a single $\mathcal L_a$ scoring $\hat s(a)=\max_{k\ge1}\hat q^{(k)}_{i_q}(a)$ against the label $a^\star$ — is a defect and is deleted: $\arg\max_a\mathrm{softmax}(-\hat s)=\arg\min_a\max_{k\ge1}q^{(k)}(a)$, which is the unconstrained risk ranking, while $a^\star=\arg\max_aq^{(0)}(a)$ is a different column, so the gradient pushed the risk ranking toward the goal ranking and neither the subject condition nor $a^\dagger$ — the column out.move returns (08.19) — appeared in any loss (D-2, M-1, V-8). Repaired: $\mathcal L_a$ scores $-\hat q^{(0)}$ against $a^\star$, so its argmax is $\arg\max_a\hat q^{(0)}(a)$, the label's own object; $\mathcal L_{a^\dagger}$ scores the constraint risk plus the feasibility hinge $\Lambda_{\mathcal F}\,\mathrm{relu}(\max_{k\ge1}\hat q^{(k)}_{i_q}(a)-\hat q^{(0)}_{i_q}(a))$ with $\Lambda_{\mathcal F}=10$ frozen [ASSUMED, restated for the hinge and not carried from the indicator: any constant exceeding the committor's range $1$ by an order makes the hinged argmin equal the constrained argmin on $\hat{\mathcal F}$, since off $\hat{\mathcal F}$ the added term is $\Lambda_{\mathcal F}$ times a positive margin bounded by $1$ and the unhinged scores it must dominate span at most $1$; to be re-registered from the realised margin sd before any cell] against the label $a^{\dagger\star}$, the constrained oracle argmin $\arg\min_{a\in\mathcal F^\star}\max_{k\ge1}q^{\star(k)}(a)$ computed on $P_{\rm env}$ with $\mathcal F^\star$ the oracle's own feasible set, and with the empty-$\mathcal F^\star$ fallback of 08.9 in force ($a^{\dagger\star}=a^\star$, infeasible=True, and the term is counted and skipped on those draws rather than trained against a fallback label).

Why the penalty is a hinge and not an indicator, printed because the indicator form trained nothing and the book said it did (D-2, M-1, V-8, V-9). The form this verse carried for one round was $\Lambda_{\mathcal F}(1-\mathbb 1[a\in\hat{\mathcal F}])$ with $\hat{\mathcal F}=\{a:\hat q^{(0)}(a)\ge\max_{k\ge1}\hat q^{(k)}(a)\}$, asserted to be "differentiable in the committors it ranks". It is not: $\partial\,\mathbb 1[\hat q^{(0)}(a)\ge\max_{k\ge1}\hat q^{(k)}(a)]/\partial\hat q=0$ for every $\hat q$ off the measure-zero set $\hat q^{(0)}(a)=\max_{k\ge1}\hat q^{(k)}(a)$, so $\Lambda_{\mathcal F}=10$ contributed exactly $0$ to $\nabla\mathcal L_{a^\dagger}$ on $100$ per cent of draws and what remained differentiable in $\hat s^\dagger$ was $\max_{k\ge1}\hat q^{(k)}$ alone — the unconstrained risk ranking, which is precisely the object the previous round's own repair identified as the wrong one ($\arg\max_a\mathrm{softmax}(-\hat s^\dagger)=\arg\min_a\max_{k\ge1}q^{(k)}$ once the penalty's gradient vanishes). The subject condition that defines the shipped column was therefore still trained by nothing, one level down from where the defect was first filed. The hinge has subgradient $-\Lambda_{\mathcal F}$ with respect to $\hat q^{(0)}(a)$ and $+\Lambda_{\mathcal F}$ with respect to the active $\max_{k\ge1}\hat q^{(k)}(a)$ on exactly the infeasible set $\{a:\max_{k\ge1}\hat q^{(k)}(a)>\hat q^{(0)}(a)\}$ and $0$ on $\hat{\mathcal F}$, and it reduces to the indicator form as $\Lambda_{\mathcal F}\to\infty$, so nothing is claimed that the old form claimed and the gradient exists where the old form's did not. a_dagger_star is a required oracle field of 08.18's table. $\mathcal L_q$ is Dirichlet-consistent in the sense that the head clamps $\hat q$ to the boundary values on $\mathcal A$ by construction (08.7) and the loss is taken on $T$ only, so the boundary condition is never learned and the interior is harmonic in $\hat P$ exactly. The harmonic residual is not a loss (D-2). Weights: $\lambda_z=\lambda_q=\lambda_\Delta=\lambda_c=\lambda_a=\lambda_{a^\dagger}=1$ with each term normalised by its label scale as written, $\lambda_{\rm LM}=1$ where present, $\tau_a=0.1$ — all [ASSUMED: equal weight after per-term normalisation is the only weighting that no cell has yet contradicted; no CEQ arm has been trained (B5)]. Schedule: stage A ($0$–$S_0$ steps) $\lambda_\Delta=\lambda_a=\lambda_{a^\dagger}=0$, $\gamma_h$ frozen at $0$ (the corner), $\mathcal L_q$ supervises $\hat P$ through the $\gamma=1$ head; stage B releases $\eta_h$, $\lambda_\Delta$, $\lambda_a$, $\lambda_{a^\dagger}$; $S_0=\lfloor S/4\rfloor$ [ASSUMED]. $i_{\min}$ is the first position after the last declared set.

Hypotheses. Labels computed on $P_{\rm env}$ (D-2); $\sigma_z$ the per-coordinate label sd over the admitted region ($>0.05$, S-12); $m_{\rm eff}=m$; 08.17's frozen set in force ($\beta_h\equiv1$, $\kappa_h\equiv1$, $g_h\equiv0$ and the two gate heads all requires_grad=False, so neither the gradient channel nor the decay channel can move them), which is 08.13's Replacement 1 promoted to 08.17's Statement — without it the decayed or the trained $\beta$ breaks row-stochasticity inside this schedule's own $150$ steps and $\mathcal L_q$ is taken against a $q$ that is not a committor (08.17). The $\Delta z$ term is refused per call, never per forward (08.19's dz_degenerate flag, V-16), and the flag is the restated predicate of 08.19 clause 1, not the one this verse carried for one round: for a clamp — the only move type BED-S registers (READ docs/CEQ_SHAPE.md:1679 @ 207e7b9, S-11: "Candidate moves are $m$ query-side row clamps") — degeneracy is $\gamma_h=0$ or $u_a^\top z=0$; for a rewrite it is $\Delta\tilde V\equiv0$ and $\Delta P\equiv0$ jointly. $\Delta\tilde V\equiv0$ alone is the normal case for a clamp and predicts nothing: a clamp changes $P$ and leaves $\tilde V$ untouched, so the old predicate held on $m/m$ moves of every draw, $\mathcal L_\Delta\equiv0$ on the bed as registered and dz_degenerate_frac read $1.000$ while 08.8's identity gives $\Delta z=\gamma(I-\gamma P')^{-1}\Delta P\,z\ne0$, measured at $0.1096$ at $s=32$ and $1.127$ at $s=64$ (08.8 Evidence, RUN[F]/RUN[J]). On a call flagged degenerate under the restated predicate — $\gamma_h=0$, which is every step of stage A and every load-parity cell, or $u_a^\top z=0$, which is the reading 08.8's Evidence actually records for $V\equiv\mathbb 1$ ($z\propto\mathbb 1$ and $u_a^\top\mathbb 1=0$, hence $\Delta z\le10^{-15}$) — both halves of $\mathcal L_\Delta$ contribute $0$ and the call is counted in dz_degenerate_frac, and the forward does not raise; the ratio half is refused there and not rescued by $\epsilon_\Delta$, whose denominator on such a call would be $\epsilon_\Delta$ alone; a raise on those calls would abort every step of stage A and every corner cell, which is a gate satisfied by construction and not a guard (V-10). On the arena at $d_{\rm model}=16$ the pair is unmatched under both counting rules of 08.15 ($1.97\times$ and $24.2\times$ Ruling 3's bar), so every number this verse's Kill reads there is a diagnostic contrast: the Kill below is a training-dynamics statement and no T1 or T2 sentence may be written from this verse at $d_{\rm model}=16$ (A1 and A2 are written in no verse of this book); the licensed geometry for T1 and T2 is the LM shape at 08.15's Rule-B count, $192$ added parameters, $0.00074627\%$, with no widening of the control's MLP anywhere — that move is deleted from 08.15 as the act Ruling 3 forbids by name.

Evidence. The label class is SHAPE §4.4's change ("no milestone scores a scalar at one position"); the metrics the losses mirror are S-16's (RUN-free; frozen before any cell, M-2); the mirror-contrast and raw-label guards of the BED-M consequence bed are inherited (SHAPE §A.1). No loss weight in this verse has a RUN number: NOT MEASURED — needs the eight-seed arena S-62 (book 04) with the weight vector in the manifest.

Mechanism. D-2 (the residual is a score); M-2 (weights frozen before the cell); V-16; D-1 (vector labels); B12 (stage A holds the corner while $\hat P$ is identified from committor labels — book 02 owns why).

Kill. After stage B, $\hat\gamma$ PINNED ($\Lambda\le2.7055$) on $\ge6$ of $8$ seeds and $\mathcal L_q$ at step $S$ not below its step-$0$ value by more than one seed sd — the objective neither moves the dial nor identifies $\hat P$; frozen. Instrument: S-61/S-62 (book 04). Price: inside the $\approx98$ s arena. Planted negatives, three, one per term whose gradient this round was shown not to exist or whose flag was shown to fire on the whole class: (i) a query token carrying $q^{\star(0)}(a^\star)$ must drive $\mathcal L_a$ to $0$ at step $0$ (the S-14 leak plant), or the loss is not reading the label; (ii) a draw whose oracle $a^{\dagger\star}$ lies outside $\hat{\mathcal F}$ must produce a non-zero gradient on $\hat q^{(0)}$ at step $0$, of magnitude $\Lambda_{\mathcal F}=10$ times the CE's own factor on that entry, or the feasibility constraint is decorative and $\mathcal L_{a^\dagger}$ is training the unconstrained risk ranking under another name (the exact reading the indicator form produced); (iii) a clamp move evaluated at $\gamma_h>0$ must read dz_degenerate=False and reproduce 08.8's measured $\Delta z$ magnitude $1.127$ at $s=64$ (RUN[J]) within the metric's own tolerance, or the flag is firing on the whole registered move class and $\mathcal L_\Delta$ is identically $0$ on every draw.

If killed. Replacement 1 — committor-only supervision. $\lambda_z=\lambda_\Delta=\lambda_a=\lambda_{a^\dagger}=\lambda_c=0$, $\lambda_q=1$, $\gamma_h\equiv0$: the module trains $\hat P$ to reproduce the reach-avoid vector through the exact $\gamma=1$ head and reads $z$ only as a diagnostic. Kill, restated on the object the identification can actually move. The full-matrix form this replacement used to carry — "$\|\hat P-P_{\rm env}\|_\infty$ not below $10^{-2}$ on $6$ of $8$" — fires on $8$ of $8$ by construction and is not a kill (V-11): BED-S has $P_{{\rm env},ii}=0$ on $T$ (READ docs/CEQ_SHAPE.md:1679, S-11) while the arm has $P_{ii}>0$ with $\max_TP_{ii}=0.6926596893360386$ (RUN[J]), so $\|\hat P-P_{\rm env}\|_\infty\ge2\max_TP_{ii}=1.3853193786720772$, which is $138.5\times$ the $10^{-2}$ bar before any step (08.12's named residual). The kill is therefore two clauses on objects the diagonal does not decide, both inside the arena, both paired: (i) $\max_{i\ne j}|\hat P_{ij}-P_{{\rm env},ij}|$ — the off-diagonal sup — not below $10^{-2}$ on $6$ of $8$ seeds; and (ii) the committor the operator induces, $\varphi$-NRMSE of $\hat q$ against $q^\star$ on the admitted region, not below its own step-$0$ value by more than $\mathrm{MDE}_8$ at the realised paired sd on $6$ of $8$ seeds. Clause (ii)'s bar is anchored in the realised sd rather than in a bare $10^{-2}$, because $10^{-2}$ on an unnormalised operator entry is a threshold with no units (V-17); clause (i)'s $10^{-2}$ is frozen against the off-diagonal's own scale, $\max_{i\ne j}P_{{\rm env},ij}\le1$. Planted negative: an arm whose $\hat P$ is set to $P_{\rm env}$ off the diagonal and to the arm's own diagonal must pass clause (i) and fail nothing, or the restriction has not been applied. Replacement 2 — interventional pairs only. $\lambda_\Delta=1$ and every other weight $0$; $\gamma_h$ learnable; the label is the displacement field alone. Kill: Bet H's counter (the cached mixture within one seed sd on $P$-changing plants), $\approx35$ s.

Terminal. $\mathcal L_{\rm LM}$ alone at the corner — a softmax LM; the canon licenses "trains as softmax trains" and withdraws every objective above.

08.17 — The optimiser regime and the record's numbers; T1/T2 as the two measured quantities#

Statement. torch.optim.AdamW over a frozen set and two parameter groups, lr $3\times10^{-4}$, betas $(0.9,0.999)$, eps $10^{-8}$, gradient clipping at global norm $1.0$, no scheduler, batch $32$ sequences (the ceq/lm.py loop) or $8$ (the HF loop at the LM shape), one RNG for data drawn on the host from seed+1 and one for evaluation from seed+2:

frozen = [beta_h, kappa_h, g_h] + list(u_head.parameters()) + list(theta_head.parameters())
for p in frozen: p.requires_grad_(False)      # BOTH channels shut: no gradient, no decay
dial   = [eta_h]                              # 08.5; clip(eta,0,1) applied after every step
rest   = [p for p in model.parameters() if p.requires_grad and p is not eta_h]
opt = torch.optim.AdamW([{"params": dial, "weight_decay": 0.0},
                         {"params": rest, "weight_decay": 0.01}],
                        lr=3e-4, betas=(0.9, 0.999), eps=1e-8)

This Statement ships 08.13's Replacement 1 and says so, rather than naming it as the alternative. With $\beta_h\equiv1$ and $\kappa_h\equiv1$ frozen, Ruling 2's "$\beta$ learnable, logged per instance" is not met: every cell of this book journals beta = frozen and weight_decay_groups = frozen_beta, and no cell claims Ruling 2 (V-22). That is a real loss and it is stated as one — the canon gives up a learnable $\beta$ to keep the row-stochasticity every read in Parts II and III is hypothesised on.

Why a frozen set and not a weight_decay: 0.0 group. DERIVED, six lines — the previous round closed the decay channel and left the gradient channel open, and the guard below then halted training at step 1 of the shipped Statement. (i) Both loops in the record decay every parameter: READ ceq/lm.py:264 @ 99777ab (opt = torch.optim.AdamW(m.parameters(), lr=lr)) and READ ceq/hf/train.py:378 @ 99777ab (opt = torch.optim.AdamW(model.parameters(), lr=lr)) — neither excludes any parameter, and AdamW's weight_decay default is $0.01$. (ii) The decay channel. Decoupled decay multiplies every parameter it touches by $(1-\mathrm{lr}\cdot\mathrm{wd})$ per step, so a learnable $\beta_h$ initialised at $1$ reads $\beta\leftarrow(1-3\times10^{-6})^S$: $\beta=0.9995501005601203$ at $S=150$ (this verse's own arena) and $\beta=0.9716107252156855$ at $S=9{,}600$ (SHAPE §5.3 M-5.3's ladder). (iii) The quantity the guard reads. At $\beta\ne1$ the operator's row sums are $Z_i^{1-\beta}$ (08.2's Statement divides by $Z_i^\beta$, and the unnormalised row sums to $Z_i$), so at $s=64$ with $Z_i\sim64$ the row-sum residual is $|1-\beta|\ln64=4.1589\,|1-\beta|$ to first order, and the decay channel alone reads $4.1589\times4.499\times10^{-4}=1.871\times10^{-3}$ at $S=150$. (iv) The gradient channel, which a weight_decay: 0.0 group does not touch at all. $\partial W/\partial\beta=-\ln Z_i\,W$ with $\ln Z_i\approx4.1589$ at $Z_i\sim64$, so the gradient on a learnable $\beta_h$ is not zero; AdamW's bias-corrected step is $\approx\mathrm{lr}\cdot\hat m/(\sqrt{\hat v}+\epsilon)$, of magnitude $\approx\mathrm{lr}=3\times10^{-4}$ for a consistently signed gradient, so one step moves $|1-\beta|$ to $3\times10^{-4}$ and the row-sum residual to $4.1589\times3\times10^{-4}=1.248\times10^{-3}$ — $327\times$ the guard's bar below, at step $1$. A Statement that kept $\beta_h$ learnable per Ruling 2 and shipped the guard would therefore halt its own training on the first optimiser step, on every seed, which is a kill that fires at every real draw (V-11 inverted); that is the state this verse was in and it is repaired by freezing, not by a decay exclusion. (v) $\kappa_h$ enters $qk_{ij}=\kappa\,q_i\cdot k_j/\sqrt d$ multiplicatively and is likewise unconstrained by any group; it is frozen with $\beta_h$. (vi) The direction of travel is the record's own: at $\hat\beta=0$ the rows sum $1.312192$ to $10.293107$ (READ V16_ARM_SMPRIME.md:28-32 @ 99777ab). $\eta_h$ stays learnable, in the weight_decay: 0.0 group for the reason 08.5 gives (decay shrinks a recovered positive $\eta_h$ back toward the clamp's lower trap), with 08.5's two-sided projection $\mathrm{clip}(\eta_h,0,1)$ after every step.

The row-sum guard, per step, in the manifest, at a bar in its own units. $\mathrm{rs}=\|\mathrm{rowsum}(P)-1\|_\infty$ on the read layer's head $0$ is computed every step and journalled as 08.22's mandatory field rowsum_residual (max over steps, and the step index at which the max occurred), beside the second mandatory field beta_final, so the two channels are separable: a run that halts with beta_final $=0.9995501$ halted through decay, one that halts with beta_final $=0.9997$ after one step halted through the gradient, and one that halts with beta_final $=1.0$ exactly halted through neither and the defect is elsewhere in the operator. Training halts at $\mathrm{rs}>\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}$ at $s=64$ — 08.3's Kill table row three, the bar on this quantity — and a halted run is journalled halt = rowsum and is not a cell. The bar is not $\epsilon_{32}$ and the previous round's guard, which froze it there, admitted runs that fire 08.3's own conservation kill: $\epsilon_{32}=1.5\times10^{-4}$ is the bar on the conservation residual of the forward substitution and its content is the amplification $|T|\cdot\|(I-Q)^{-1}\|_\infty=1{,}500$; a row-sum residual $\eta$ propagates as $\sum_\bullet q=\mathbb 1+(I-Q)^{-1}\eta$, so $\eta=\epsilon_{32}$ yields a conservation residual of $23.81\times1.5\times10^{-4}=3.5715\times10^{-3}$, $23.81\times$ 08.3's float32 bar and $3.57\times10^{5}$ times its float64 bar — a cell passing the old guard at all $150$ steps and failing 08.3's and 08.7's kill on the same operator (DERIVED, 08.3's third Kill row). At $\epsilon_{\rm rs}$ the induced conservation residual is $\le9.083\times10^{-5}=0.606\times\epsilon_{32}$, inside. The bar is re-derived at any other $s$ (V-22). The guard costs one $s$-vector reduction per step ($s^2$ adds, $4{,}096$ at $s=64$, below the clock's resolution) and it is the instrument that decides whether the frozen set was in force, so it is mandatory on every cell.

The arena cell: steps=150, $n_{\rm train}=2048$, $n_{\rm eval}=4096$, $s=64$, $d_{\rm model}=16$, seeds $0$–$7$, one thread lane (threads=8), the flag regime on the header. The two licensed training numbers: T1 GPU-seconds from initialisation to a fixed distance-to-floor, and T2 draws to the same point, both against base self-attention at matched parameters and against the depth skyline $\lfloor\log_2t^\star\rfloor+2=5$ at $t^\star=8$, with synchronised timers, interleaved arms and run order recorded. Neither exists: NOT MEASURED — needs the T1/T2 protocol of book 04 on the cost-law instrument of book 03 (S-66's two synchronize() calls and randomised order).

Hypotheses. Ruling 1's regime (08.22); the training-noise floor (two identical-seed chunks, $|\Delta|$ final loss) measured once before any T1 ratio is read — NOT MEASURED (READ COSTS.md §4 @ 99777ab, "☐ NOT YET MEASURED"); the parity run's lr $10^{-3}$ at $600$ steps is a different regime and is not carried (V-22); the frozen set of the Statement in force for every cell — a cell run on the ungrouped optimiser, or on a weight_decay: 0.0 group with $\beta_h$ still learnable, is journalled and refused, not repaired after the fact, and beta_final is what a reader checks. Geometry, printed because T1 and T2 are the two forms 08.15 restricts: the arena at $d_{\rm model}=16$ is not a matched pair under either counting rule — $0.062906\%$ ($1.97\times$ Ruling 3's $0.032\%$ bar) under the shipped Rule B and $0.775844\%$ ($24.2\times$) under Rule L — and no move closes it, the bar at that geometry being $0.032\%\times4{,}769=1.53$ parameters, finer than one switch (08.15). T1 and T2 may therefore not be written at $d_{\rm model}=16$; the arena is a training-dynamics cell only, and the licensed geometry for T1/T2 is the LM shape ($d_{\rm model}=512$, $h=8$, $L=8$) at 08.15's Rule-B count — $192$ added parameters, $25{,}728{,}192$ against $25{,}728{,}000$, residual $0.00074627\%$, $42.9\times$ inside Ruling 3's bar, with no widening of the control's MLP, that move having been deleted from 08.15 as the act Ruling 3 forbids by name at $5.47\times$ its stated size. A1 and A2 are written in no verse of this book at any geometry (Preface item 6; 06_PREDICTIONS.md:302). The Kill below is stated at that geometry and its price is repriced from the arena rate accordingly.

Evidence. READ ceq/lm.py::train_one (lr=3e-4, bs=32, clip_grad_norm_ 1.0), ceq/hf/train.py::train (lr=3e-4, clip=1.0, AdamW, no scheduler — the file says "there is no scheduler"), @ 99777ab; the arena settings SHAPE §5.3 S-62; the record's timer hazard: no torch.cuda.synchronize in scripts/v15_r1.py (RUN grep this session: $0$ hits) and run order the strongest correlate of seconds, $\rho=+0.7029$, $p=0.0024$ (READ V20_R15_JOURNAL.md:53 @ 99777ab, C17); the per-cell prices $1.680$ s (shape, [FITTED+RUN]) and $1.524$ s (control, [FITTED]) at $n=2048,s=64$ (SHAPE §A.11 item 5).

Mechanism. B25 (never defined, never measured); C17; M-8; V-22.

Kill, in two clauses at two geometries, because one of them is the only one at which T1 and T2 are licensed. Clause 1 (the row-sum guard, the arena, $0$ GPU-s on top of it): rowsum_residual above $\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}$ at $s=64$ at any step of any of the eight arena seeds. The inference the clause licenses, restated, because the one it used to license could not distinguish the defect it named from the Statement working as specified: what a reading above $\epsilon_{\rm rs}$ says is that the operator is not row-stochastic, and $\|\mathrm{rowsum}(P)-1\|_\infty$ alone does not say through which channel — a grouped optimiser with a learnable $\beta_h$ and an ungrouped one produce the identical reading. The clause therefore reads rowsum_residual beside beta_final: beta_final $\ne1.0$ names the $\beta$ channel and beta_final $=1.0$ with rowsum_residual above the bar names the operator itself. The old inference — "the optimiser is then not running the parameter groups above" — is deleted. On any reading above the bar every read of Parts II and III is void on that cell; the run halts, is journalled halt = rowsum, and no T1, T2 or capability number is taken from it. Frozen at $\epsilon_{\rm rs}$, re-derived at any other $s$ (V-22); instrument: the per-step reduction of the Statement, journalled in 08.22. Clause 2 (T1/T2, the LM shape only): T1 read on the certified device with the timer hazards removed, on the bed named in the row and at the matched count printed in it, because CHARTER §1 requires the bed, the floor, $N$ and the matched count in the same row as the T1/T2 sentence and this clause used to name only the first two — the bed is BED-S as registered at $t^\star=8$, $K=2$, $m=8$ (SHAPE §5.3 S-11, S-62; book 04 owns the registration), the geometry is $d_{\rm model}=512$, $h=8$, $L=8$ with no change of any kind to the control, the matched count is $25{,}728{,}192$ (shape) against $25{,}728{,}000$ (control), residual $192$ parameters $=0.00074627\%$, $42.9\times$ inside Ruling 3's $0.032\%$ under the Rule-B counting rule the cell journals (08.15's closed count; under Rule L the same pair at per-layer switch granularity reads $25{,}736{,}232$ against $25{,}728{,}000$, $0.031996\%$, matched by $1.00012\times$, and the cell journals which rule it used), $N=8$ seeds $0$–$7$, and the depth skyline $\lfloor\log_2t^\star\rfloor+2=5$ is read in the same row: the shape's GPU-seconds-to-floor above base self-attention's at the same floor on $6$ of $8$ seeds, at $\lambda$ as in 08.16 — "faster to train" is then false at depth 1 and the T1 sentence is withdrawn; T2 likewise on draws. Frozen: the floor is the exact restricted-view floor of the bed's book, the same for both arms; the ratio's interval must exclude $1$. Instrument: book 04's T1/T2 protocol. Price: NOT MEASURED — needs the LM-shape step rate on the certified device (scripts/k_cert.py, book 05); the arena's $\approx26$ s and the skyline's $\approx64.8$ s are $d_{\rm model}=16$ rates and may not be carried to $d_{\rm model}=512$ (V-22, M-8), and the record's only LM-shape projection is $132{,}430$ steps per $11$ h chunk for the Q3 shape ([FITTED], READ COSTS.md §3.2 @ 99777ab), which prices the cell in hours and not in seconds. Planted negatives, three, one per channel the guard must separate plus one for the timer: an arm run with weight_decay=0.01 on all parameters must drive rowsum_residual above $\epsilon_{\rm rs}$ within $150$ steps at $s=64$ with beta_final $=0.9995501005601203$ (the DERIVED prediction is $1.871\times10^{-3}$, $490\times$ the bar), or the decay channel is invisible to the guard; an arm run with $\beta_h$ learnable and undecayed (the weight_decay: 0.0 group this verse used to ship) must drive rowsum_residual above $\epsilon_{\rm rs}$ at step $1$ with beta_final $\ne1.0$ (the DERIVED prediction is $1.248\times10^{-3}$, $327\times$ the bar), or the gradient channel is invisible and the previous round's repair would have passed its own guard; and the arm at $\gamma\equiv0$ must read a T1 ratio of $1.00$ within the clock's $\pm12\%$ against itself under the same harness, or the timer is not paired.

If killed. Both replacements inherit the frozen set of the Statement and the guard at $\epsilon_{\rm rs}$; a replacement that inherited a learnable $\beta_h$ would inherit both the halt-at-step-1 strike and the unit defect in the bar, and would not be a replacement (V-9). Replacement 1 — T2 only. Draws-to-floor is claimed and GPU-seconds withdrawn: "sample-efficient, not wall-clock-efficient". Kill: T2's ratio interval containing $1$ at the LM shape — the same cell, no timer at all, so the clock's $\pm12\%$ and the $2.0\times$–$6.6\times$ dispatch band leave the number untouched; strictly more decisive on the remaining claim and strictly cheaper because it needs no synchronisation harness. Replacement 2 — the per-step cost law, a different number on a different object measured by a different instrument. No training claim of any kind survives here: what is stated is the DERIVED MAC increment of 08.20 at the geometry this verse's own Hypotheses license, $\mathbf{1.517578\times}$ the head at the LM shape $(s,d,K,m,h)=(512,64,2,8,8)$ — $203{,}685{,}888$ MACs per layer against the softmax layer's $134{,}217{,}728$ — with $2.125\times$ named as the $h=1$ design-geometry reading at $(64,16,2,8,1)$ and not carried to the LM shape (M-8, V-22), against the K-9 $s$-sweep, an untrained forward-only measurement on the $s$-ladder $(s,n)\in\{(64,2048),(256,2048),(1024,128),(4096,8)\}$ with no optimiser, no seeds-to-floor and no arena. Kill: K-9 (08.20) — the fitted exponent's interval in $s$ excluding $2$ toward $3$, or the solve-to-product ratio above $3$ at any $s\le1024$; price $206$–$537$ GPU-s (band). It is the strictly more decisive link because it removes the optimiser from the chain entirely, and no weight-decay, parameter-group or row-sum defect can reach it.

Terminal. The module trains as a softmax stack trains, at $+50\%$ causal MACs on the $z$ channel and, for the read layer's total, $1.517578\times$ the head at the LM shape $(s,d,K,m,h)=(512,64,2,8,8)$ — the only geometry at which this verse licenses T1 and T2 — with $2.125\times$ the head at $(64,16,2,8,1)$ printed as the $h=1$ design-geometry reading and not carried across geometries (DERIVED, per-op floor; 08.6's If-killed and 08.20's two-block table print all three figures and this Terminal quotes the licensed one first). The ratio falls from $2.125\times$ to $1.517578\times$ because the $K+2$ indicator channels, the committor head and the $m$ move columns exist on head $0$ only (08.1's Head convention, 08.14's Statement) and are therefore amortised over $h=8$ heads, which the previous round's per-head table priced $8$ times over. The canon withdraws "faster to train" in both forms.

Part VI — The data interface#

08.18 — What a task supplies: tokens, sets, interventions, labels#

Statement. One task record per sequence:

field	shape / type	required	source
`tokens`	`int64[s]` (byte-level, vocab $256$, or the bed's node tokens as multi-hot features `float32[s, d_in]`)	yes	task
`masks`	`bool[K+2, s]`, channel order (sink, goal, $\mathcal A_1..\mathcal A_K$)	yes; $K\ge2$	task
`i_q`	`int64`	yes; $i_q\in T$, every set $<i_q$	task
`moves`	list of $m$ `("clamp", v_a, u_a)` with $u_a\in T$, $v_a\le i_q$ reachable, or `("rewrite", i, tokens_i')`	for $\Delta z$, $a^\dagger$	task
`gamma_env`	`float32` in $[0,1)$	for the discounted label only	task
`z_star`	`float32[s, d]` on $P_{\rm env}$ at `gamma_env`	for $\mathcal L_z$	oracle on $P_{\rm env}$
`q_star`	`float32[K+2, s]` at $\gamma=1$ on $P_{\rm env}$'s transient block	for $\mathcal L_q$	oracle (`bed_1.committor`'s form)
`dz_star`	`float32[m, s, d]` by the displacement identity on $P_{\rm env}$	for $\mathcal L_\Delta$	oracle
`a_star`, `y`	`int64`, `bool[m, K]`	for $\mathcal L_a$, $\mathcal L_c$	oracle: $a^\star=\arg\max_aq^{\star(0)}(a)$ on $P_{\rm env}$
`a_dagger_star`, `infeasible_star`	`int64`, `bool`	for $\mathcal L_{a^\dagger}$	oracle: the constrained argmin $\arg\min_{a\in\mathcal F^\star}\max_{k\ge1}q^{\star(k)}(a)$ on $P_{\rm env}$, $\mathcal F^\star=\{a:q^{\star(0)}(a)\ge\max_{k\ge1}q^{\star(k)}(a)\}$; on $\mathcal F^\star=\emptyset$ it is $a^\star$ with `infeasible_star=True` and the term is skipped and counted (08.9, 08.16)
`seed`, `generator_id`, `sha256`	Ruling 7 triple	yes	task

The oracle never sees $\hat P$; the arm never sees $P_{\rm env}$ except through the edge tokens (leak clause (c), S-15). Worked example 1 — chess (book 09): tokens are FEN-derived piece-square bytes per ply; $\mathcal A_0$ = positions whose evaluation-sign or mate-in-$k$ flag marks the side-to-move's winning class, $\mathcal A_1$ = material-loss positions, $\mathcal A_2$ = mate-against positions, the game's terminal outcome as the absorbing class; a move is ("rewrite", s-1, next-FEN tokens) — rank $s-i$, priced as a suffix re-solve; labels from a chess engine as the exact oracle where one exists; data from local lichess dumps or python-chess-generated games hashed per Ruling 7 before any Kaggle attach (the three lichess_*/tinystories_cdla sources are UNPINNED, READ COSTS.md §0 @ 99777ab). Worked example 2 — prediction markets (book 09): tokens are order-book and price-path bytes; the contracts' resolutions are the absorbing sets ($\mathcal A_0$ = resolves YES, $\mathcal A_1$ = resolves NO, $\mathcal A_2$ = drawdown breach, $\mathcal A_3$ = liquidity breach); $q^{(0)}$ is the probability of resolving in $\mathcal A_0$ before any other set; a trade is a clamp on the trader's own row; the honest floor is the market-implied probability and the counter is near-efficiency.

Hypotheses. F1/F2 hold for the task's masks; the task's oracle is exact or its own error is printed; look-ahead is excluded by construction (book 09's leakage guards).

Evidence. The bed specification SHAPE §5.3 S-11 (the DAG environment, query-side clamps, the label tensor); ceq/kdata.py carries label_plies, fen_dedupe, split_by_game (READ ceq/kdata.py:206-320 @ 99777ab) — plumbing for the chess route, no cell; Ruling 7 (READ V17K_RULINGS.md:83-86 @ 99777ab).

Mechanism. D-2; V-22 (E4′'s bars re-derived, never carried); M-1; B27 (owned by book 09).

Kill. S-15 frozen: the corpus-alone linear probe from token features to $q^\star$ at order $0$ reads $R^2\ge0.5$ on the admitted draws — the one-read claim is void on that task; or the planted leak (a move token carrying its own $q$) reads $R^2<0.99$ — the detector is blind and no leak verdict is admissible. Price: $0$ GPU-s (CPU probes). Instrument: book 04 S-15; for chess and markets, book 09's leakage guard.

If killed. Replacement 1 — strip the leaking features (membership flags removed from the token features; sets enter only through masks) and re-probe; kill: $R^2$ still $\ge0.5$, $0$ GPU-s. Replacement 2 — the task is a copy task and is struck (S-14); what survives is the interface table above with no label, which is what a task must supply to be admitted.

Terminal. The interface is a specification without an admitted task; the canon licenses the field table and withdraws every capability sentence on the task.

Part VII — The inference API#

08.19 — One forward call, and what it returns#

Statement.

out = ceq.forward(tokens, masks, i_q, moves=None, gamma_env=None, return_moves=True)
# out.O        float32[B, s, d_model]     residual-stream output (08.6)
# out.z        float32[B, h, s, d+K+2]    next transient state, all channels (08.4)
# out.q        float32[B, K+2, s]         committor vector, exact at gamma=1 (08.7)
# out.dz       float32[B, m, s, d]        displacement per move, None if moves is None (08.8)
# out.dz_degenerate  bool[B, m]           True where dz is an ANALYTICALLY PREDICTED zero (08.8, 08.16)
# out.move     int64[B]  (a_dagger),  out.move_cols {a_star, a_dagger, a_ddagger, lex}: int64[B] each,
#                                     plus infeasible: bool[B]  and  s_discarded: bool[B, m]   (08.9)
# out.phase    (phi: int64[B], margin: float32[B], tau: float32[B], absorbed: bool[B]) (08.10)
# out.cert     (delta: float32[h], V_inf: float32[B], one_over_1mg: float32[h], route: str) (08.11)
# out.gamma    float32[h]; out.beta float32[h]   (Ruling 2: logged per instance)

Batch rule: $B$ sequences share $s$, $K$, $m$; ragged $K$ or $m$ are padded with an all-zero mask channel or a no-op move and the padding is counted in $m_{\rm eff}$ (R-08). A call with i_q in $\mathcal A$ returns absorbed=True and reads no solve.

Which column ships and which loss trains it, with the differentiable term named. out.move returns $a^\dagger$, the constrained column of 08.9, and $\mathcal L_{a^\dagger}$ of 08.16 — score $\max_{k\ge1}\hat q^{(k)}_{i_q}(a)$ plus the hinge $\Lambda_{\mathcal F}\,\mathrm{relu}(\max_{k\ge1}\hat q^{(k)}_{i_q}(a)-\hat q^{(0)}_{i_q}(a))$, label the constrained oracle argmin a_dagger_star of 08.18 — is the loss that trains it. The hinge is what makes that sentence true: it carries subgradient $\Lambda_{\mathcal F}$ on exactly the infeasible set, whereas the indicator form $\Lambda_{\mathcal F}(1-\mathbb 1[a\in\hat{\mathcal F}])$ this line asserted for one round has gradient identically $0$ in $\hat q$, so the shipped column was trained by the unconstrained risk ranking and this line was false while it stood (08.16). $\mathcal L_a$ (score $-\hat q^{(0)}$, label a_star) trains out.move_cols.a_star and no other column.

The three raises and the two flags, separated, because a raise on an analytically predicted zero fires by construction. The rule this verse used to carry — "a call whose dz is identically $0$ on any sequence raises" — is a gate satisfied by construction at the one corner the module is required to load and to train stage A at (V-10, V-16 misapplied): 08.8's identity is $\Delta z(a)=\gamma c_au_a^\top z/(1-\gamma u_a^\top c_a)$, which is identically $0$ for every move at $\gamma=0$, and 08.13's load rule sets $\gamma_h=0$ bitwise at load while 08.16's stage A freezes $\gamma_h$ at $0$ for $S_0=\lfloor S/4\rfloor=37$ of the arena's $150$ steps — so with return_moves=True the default, every forward carrying moves raised for the whole of stage A and at every load-parity cell. 08.8's Evidence records the same predicted zero for constant values ($V\equiv\mathbb 1\Rightarrow\Delta z\le10^{-15}$). The repaired rule, three outcomes and never a silent number:

Predicted zero, flagged, returned — on the predicate the identity actually supports, per move type. For a clamp (08.8(i)), the analytic zeros are $\gamma_h=0$ or $u_a^\top z=0$; the second is $O(s)$ to test on the already-computed $z$, runs before the solve of the column $c_a$, and is what 08.8's Evidence records for $V\equiv\mathbb 1$ ($z\propto\mathbb 1$ and $u_a^\top\mathbb 1=0$, hence $\Delta z\le10^{-15}$ — the zero comes from $u_a^\top z=0$, not from any property of $\tilde V$). For a rewrite (08.8(ii)), the analytic zero is $\Delta\tilde V\equiv0$ and $\Delta P\equiv0$ jointly. The call returns $\Delta z=0$ with dz_degenerate[b,a]=True, $\mathcal L_\Delta$ contributes $0$ on those entries, they are counted in dz_degenerate_frac (08.16, 08.22), and the forward does not raise. The predicate this verse carried for one round — "$\gamma_h=0$ or $\Delta\tilde V\equiv0$" — is deleted, because $\Delta\tilde V\equiv0$ alone is the normal case for a clamp and predicts nothing: a query-side row clamp changes $P$ and leaves $\tilde V$ untouched, and a clamp is the only move type BED-S registers (READ docs/CEQ_SHAPE.md:1679 @ 207e7b9, S-11: "Candidate moves are $m$ query-side row clamps $P_{\rm env}[v_a,:]\leftarrow e_{u_a}$"), so the flag held on $m/m$ moves of every draw, out.dz was returned as $0$ against 08.8's measured $1.127$ at $s=64$ (RUN[J]), 08.16 refused both halves of $\mathcal L_\Delta$ on every entry, $\mathcal L_\Delta\equiv0$ and dz_degenerate_frac read $1.000$ on the bed as registered — an analytic-zero gate satisfied by construction on the whole move class (V-10, V-16, D-2), which is the mirror of the by-construction raise it replaced.
Unpredicted zero, raises. When the restated predicate of clause 1 is False — for a clamp, $\gamma_h>0$ and $u_a^\top z\ne0$, so the identity predicts a non-zero magnitude $|\gamma c_au_a^\top z/(1-\gamma u_a^\top c_a)|>0$; for a rewrite, $\Delta\tilde V\not\equiv0$ or $\Delta P\not\equiv0$ — and the returned $\Delta z$ is numerically zero, the call raises: the cosine is undefined and nothing analytic explains it. This is the outcome the old rule was reaching for and the only one it should ever have fired on.
The Chebyshev bar, raised or discarded per move. A move with $q^{(\rm sink)}(a)>1-\epsilon_{\rm sink}$, $\epsilon_{\rm sink}=10^{-3}$ frozen (08.9), is discarded from $\mathcal S$, flagged s_discarded[b,a]=True and counted, never divided; $a^\ddagger$ and 08.10's conditioned margin are computed on the survivors. A call in which every move is discarded raises rather than returning an $\arg\min$ over an empty set. A batch whose $\mathcal F$ is empty returns $a^\dagger=a^\star$ with infeasible[b]=True (08.9's frozen fallback) and does not raise, because that outcome is defined.

Hypotheses. 08.1–08.11; $\gamma_h$, $u_a$, $z$ and $\Delta\tilde V$ readable before the column solve of $c_a$, which is what makes clause 1's prediction analytic rather than a post-hoc excuse — $u_a^\top z$ costs $O(s)$ on the already-computed $z$, and $\Delta P\equiv0$ is a mask comparison; the move type (clamp or rewrite) is on the moves argument (08.8), so the predicate branches on a declared field and not on a measurement.

Evidence. The record's forward returns [n] (READ ceq/arm_smprime.py::ArmSMPrime.forward @ 99777ab) — the single-location shape of D-1; this API is its replacement and has no FOUND record: NOT MEASURED — needs M-2.0 (the arm as a registered kind, book 04/05).

Mechanism. FOUND-not-NAMED; V-16; D-1.

Kill. At the end of Phase 2 (SHAPE §5.3 M-2.0) results/ holds zero records with the arm's kind — the API is NAMED and struck (the R15 kill applied to this lane). Price: $0$ GPU-s (a census by kind, calibrated both ways: arm_phase must read $0$ and softmax above $0$, RUN census $191/182/182/0$ at SHAPE §4.4).

If killed. The two-head API of 08.1's replacement (H-q, H-z) with the same census kill — identical price, and decisive because it is the smallest surface that can be FOUND. It carries clause 1's predicate as restated above and not as it was: (H-z) returns $\Delta z$ per move with dz_degenerate flagged on $\gamma_h=0$ or $u_a^\top z=0$ for a clamp and on $\Delta\tilde V\equiv0$ and $\Delta P\equiv0$ jointly for a rewrite; a two-head API that inherited the "$\Delta\tilde V\equiv0$" predicate would inherit the flag firing on $m/m$ of every draw and would not be a replacement (V-9).

Terminal. forward(tokens) -> logits; a softmax LM.

Part VIII — Budget#

08.20 — The per-layer, per-step FLOP and byte model on the certified RTX 4060 Laptop and on a Kaggle T4/P100#

Statement, in two blocks, because three of the seven rows exist on head $0$ only and pricing them on all $h$ heads over-counts the layer by $8\times$ on those rows at the LM shape (M-8, V-22). After 08.1's Head convention and 08.14's Statement, the $K+2$ indicator channels, the committor head and the $m$ move columns are carried by head $0$ alone; heads $1..h-1$ run at $\mathcal A=\{0\}$ and carry the softmax head and the $d$-channel solve and nothing else. MACs per sequence (DERIVED; causal halves counted as the record counts them, SHAPE §2.2 Prop. 5(a)):

Block 1 — per head, all $h$ heads:

stage	MACs	at $s=64,d=16$	at LM $s=512,d=64$	depth
softmax head ($QK^\top$ causal $+\,PV$ causal)	$s^2d$	$65{,}536$	$16{,}777{,}216$	1
gate path product (only when $g\ne0$; never in the shipped module)	$s^2/2$	$2{,}048$	$131{,}072$	$s$ (cumprod)
the solve, $d$ value channels	$s^2d/2$	$32{,}768$	$8{,}388{,}608$	$s$
per-head subtotal (shipped, $g\equiv0$)	$\tfrac32s^2d$	$98{,}304$	$25{,}165{,}824$	$s$

Block 2 — per layer, head $0$ only, once regardless of $h$:

stage	MACs	at $s=64,d=16,K=2,m=8$	at LM $s=512,d=64,K=2,m=8$	depth
the solve, $K+2$ indicator channels	$s^2(K+2)/2$	$8{,}192$	$524{,}288$	$s$
committor head at $\gamma=1$ on $T$	$	T	^2(K+2)/2\le s^2(K+2)/2$	$\le8{,}192$
$m$ clamp moves (column route)	$m\,s^2/2+m\,s\,d$	$16{,}384+8{,}192$	$1{,}048{,}576+262{,}144$	$s$
head-$0$ subtotal	$s^2(K+2)+ms^2/2+msd$	$40{,}960$	$2{,}359{,}296$	$s$

Layer total $=h\times$ (Block 1 subtotal) $+$ (Block 2 subtotal), against the softmax layer's $h\,s^2d$:

geometry	layer total	softmax layer	ratio
design, $(s,d,K,m,h)=(64,16,2,8,1)$	$98{,}304+40{,}960=139{,}264$	$65{,}536$	$\mathbf{2.125\times}$
LM shape, $(s,d,K,m,h)=(512,64,2,8,8)$	$8\times25{,}165{,}824+2{,}359{,}296=203{,}685{,}888$	$8\times16{,}777{,}216=134{,}217{,}728$	$\mathbf{1.517578\times}$

The $2.125\times$ figure is an $h=1$, $d_{\rm model}=16$ reading and may not be quoted at the LM shape (V-22, M-8); the LM shape is the only geometry at which 08.15 and 08.17 license T1 and T2, so $1.517578\times$ is the figure 08.17's Terminal and its Replacement 2 quote, with $2.125\times$ named beside it as the design-geometry reading. The previous round's single table headed "Per head, per sequence" totalled $139{,}264$ with all three head-$0$-only rows inside it and exported $2.125\times$ to a verse whose only licensed geometry is $h=8$, $d_{\rm model}=512$. For the full sequence batch the per-step cost is $n\times$(layer total) forward, $\approx2\times$ for the backward. Bytes retained per head per sequence, per operator dtype, because the record's operator and this book's shipped shape are not the same dtype (08.21 Hypotheses; M-8): $P$ at $4\,s^2$ B on the shipped real path at $g\equiv0$ ($16$ KiB at $s=64$, $64$ MiB at $s=4096$) and at $8\,s^2$ B on the record's arm_smprime complex64 path ($32$ KiB and $128$ MiB at the same two points; RUN results/k_cert_local.json, "operator_bytes_per_element": 8), $z$ at $4\,sd$ B per head for the value channels plus $4\,s(K+2)$ B once per layer for head $0$'s indicator channels — not $4\,s(d+K+2)$ B per head, which prices the indicator channels $h$ times (08.1's Head convention; 08.21 carries the same correction into its residency delta) — and $M=I-\gamma P$ materialised or fused (book 03). Measured on the certified device (float32, $n=2048$, $s=64$, $d=16$, fwd+bwd, per-op floor): solve $+\,Pz$ $2.514$ ms vs $PV$ $1.473$ ms vs one Neumann hop $3.001$ ms; increments $1.041/2.312/4.542$ ms at $n=2048/4096/8192$ (RUN NEPTUNE). The whole-step law on the softmax corner: s/step = exp(−12.1852)·n^0.9963, $R^2=0.999998$ ([FITTED], READ COSTS.md §1.1 @ 99777ab), i.e. $0.010142$ s at $n=2048$ ([MEASURED], results/k_cert_local.json); the corner-3 base (arm_smprime, complex64) is $9.2\times$ dearer at $n=2048$ ($0.093052$ s, [MEASURED]) and is not the shipped base. Kaggle: the T4-16GB budget is $15.0\times0.90=13.50$ GiB [MODULE ceq/sizing.py]; the P100 and T4 certificates are EMPTY (READ COSTS.md §2 @ 99777ab) and every Kaggle step rate is NOT MEASURED — needs scripts/k_cert.py --out results/k_cert_kaggle.json on the card (book 05); the local law projects $425{,}506$ steps per $11$ h chunk for arm_smprime at $n=2048$ and $132{,}430$ for the LM Q3 shape ([FITTED]/[MEASURED], READ COSTS.md §3.2 @ 99777ab), and a T4 number may not be pooled with a local one (V-22). Device peak FLOP rates are not in the record and are not asserted.

Hypotheses. The MAC model is arithmetic; the dispatch gap of $2.0\times$–$6.6\times$ between the FLOP model and wall-clock (READ scale/m3_flops.py:101-121 @ 99777ab) applies until S-66 runs; the clock is non-stationary at $\pm12\%$; the certificate is stale (torch 2.5.1+cu121 at ab5b485; the box runs 2.14.0, B1/B2 — book 05).

Evidence. As tagged per number above; SHAPE §2.5; COSTS.md §1.1, §1.3, §3.2.

Mechanism. M-8 (increments over the contained head; each arm at its own rate); P-8 (a floor stated as a floor); B14; B15; C17.

Kill. K-9 (SHAPE §5.3 S-66; book 03): on the synchronised, order-randomised $s$-sweep at $(s,n)\in\{(64,2048),(256,2048),(1024,128),(4096,8)\}$, $N=8$, the fitted exponent's interval in $s$ excludes $2$ toward $3$, or the solve-to-product ratio exceeds $3$ at any $s\le1024$ (N-09). Price: the record's band $206$–$537$ GPU-s, band only. Planted negative: the dense per-op microbenchmark must reproduce $1.041$ ms within $\pm12\%$ at $n=2048$ before the sweep is read (N-02), or the harness is not the one the reference was taken on.

If killed. Replacement 1 — the chunked block-triangular solve (book 03 N-11): $M_{kk}z_k=\tilde v_k-\gamma\sum_{l<k}P_{kl}z_l$, $s/C$ diagonal solves, $\approx s^2d/2+sCd/2+sC^2/3$ MACs, depth $s/C$, retained blocks $sC$; parity $10^{-6}$ against the serial solve. Kill: slower at every $C\in\{32,64,128\}$ at $s=4096$, or parity fails — NOT MEASURED — needs the chunked kernel; first timing minutes once built. Replacement 2 — dense per chunk with $n$ declared per $s$ (N-20's residency table): the serial route, batch reduced until reserved bytes $<7.996$ GiB. Kill: reserved above $7.996$ GiB at $n=16384$, $s=64$ under a training loop (the arm's own reading was $10.578$ GiB, [MEASURED]); $\approx3$ GPU-min.

Terminal. The MAC law as a MAC law, measured points quoted as points; the canon withdraws every $s$-scaled price and the "plus 50 per cent" wall-clock sentence (SHAPE §5.10 S-66 row).

08.21 — Memory residency of the shape at the softmax corner, and the $n$ per $s$ rule#

Statement, with every memory number in GiB against a card stated in GiB, because the same figure read in GB and printed as GiB inverts the residency verdict (V-17, V-22, P-8). The certificate stores bytes; the card is $8{,}585{,}216{,}000$ B $=7.996$ GiB $=8.585$ GB (total_vram, RUN python -c "import json;d=json.load(open('results/k_cert_local.json'))" on memory.cells @ 99777ab). Allocated/reserved for one forward+backward at $s=64$, converted at $2^{30}$:

kind	$n$	peak (B)	reserved (B)	peak/reserved (GiB)	`resident`
`softmax`	$2{,}048$	$302{,}270{,}976$	$379{,}584{,}512$	$0.281/0.354$	`true`
`softmax`	$32{,}768$	$4{,}328{,}958{,}464$	$5{,}270{,}142{,}976$	$4.032/4.908$	`true`
`arm_smprime`	$2{,}048$	$874{,}290{,}688$	$1{,}061{,}158{,}912$	$0.814/0.988$	`true`
`arm_smprime`	$16{,}384$	$6{,}759{,}747{,}072$	$8{,}057{,}257{,}984$	$6.296/7.504$	`true`
`arm_smprime`	$32{,}768$	$13{,}484{,}888{,}576$	$14{,}979{,}956{,}736$	$12.559/13.951$	`false`

softmax is resident throughout, to $4.908$ GiB reserved at $n=32{,}768$ — the same reading 03_KERNEL.md:131 prints from the same cells. The residency verdict at arm_smprime, $n=16{,}384$, restated: reserved is $7.504$ GiB, which is below the card's $7.996$ GiB, and the certificate's own field reads "resident": true — so the complex corner-3 base is resident at $n=16{,}384$ outside a training loop. The sentence this verse used to print, that $8.057$ was above the card, compared the figure in GB against a card in GiB and concluded non-residency against the certificate's own field and against book 03's reading of the identical cells; it is deleted. What is true and is kept: under a training loop the same shape reads $10.578$ GiB (READ COSTS.md §1.3 @ 99777ab), which is $1.323\times$ the card and not resident — that is Ruling 8's Q2 drop, and it is a training-loop number, never the certificate's forward+backward number. At $n=32{,}768$ the base is not resident on either reading ($13.951$ GiB reserved, "resident": false). The shape on the softmax corner inherits the softmax rows plus, for $z$ and the solve's transients, $4sd$ B per head per sequence for the value channels and $4s(K+2)$ B once per layer per sequence for head $0$'s indicator channels — the two groups scale differently in $h$ and are therefore priced apart (08.1's Head convention, 08.14's Statement; M-8): the $K+2$ indicator channels, the committor head and the move columns exist on head $0$ alone, so pricing $z$ at $4s(d+K+2)$ B per head, as this verse did for one round, over-counts the indicator part by $h$ and inflates the delta $8\times$ at the LM shape. Two increments, printed apart, with the delta between them named (V-17, M-8, P-8): the $d=16$ value channels alone cost $2048\times64\times16\times4=8{,}388{,}608$ B $=+8.0$ MiB per $2048$ sequences per head, which is exactly SHAPE §5.3 N-20's figure (READ docs/CEQ_SHAPE.md:2084-2090 @ 99777ab) and is the increment the record's own arm would carry; the $K+2=4$ indicator channels of 08.4 cost $2048\times64\times4\times4=2{,}097{,}152$ B $=+2.0$ MiB once per layer, DERIVED, so at the design geometry ($h=1$) the layer's $z$ increment is $+10.0$ MiB $=10.486$ MB and at $h=8$ it is $8\times8.0+2.0=+66.0$ MiB and not $8\times10.0=+80.0$ MiB — a $21$ per cent over-count on the increment, which is what the per-head form produced. The sentence this verse used to print bridged the two with an "i.e." — "$2048\times64\times20\times4=10.5$ MB, i.e. $+8$ MiB … as N-20 states" — which equates $10.0$ MiB with $8.0$ MiB, a $25$ per cent error made by quoting a $20$-channel arithmetic and a $16$-channel citation as one number; the "i.e." is deleted and neither figure is carried into the other's accounting. Rule: $n$ is declared per $s$ so that projected reserved bytes $\le0.90\times$ the card ($0.90\times7.996=7.196$ GiB local; $13.50$ GiB T4), using the worst measured reserved/allocated ratio over resident shapes, $1.256$ ([MEASURED], READ COSTS.md §3.2) — the ratio and the $n$-per-$s$ rule are dimensionless and are unaffected by the unit defect above. At $s=4096$ two residency rules are printed, one per operator dtype, because the record's own operator is not the one this book ships and a rule stated in the wrong dtype halves or doubles the admitted $n$ (M-8, V-22, B15). Real path (this book's shipped shape, $4$ B/element): the dense $P$ alone is $4096^2\times4=67{,}108{,}864$ B $=64$ MiB per head per sequence, so $n\le\lfloor7.196\,\mathrm{GiB}/(1.256\times64\,\mathrm{MiB}\times3)\rfloor=30$ with $P$, $M$ and the gradient of $P$ retained. Complex path (the record's arm_smprime, $8$ B/element): $P$ is $128$ MiB per head per sequence and the same rule reads $n\le\lfloor7.196\,\mathrm{GiB}/(1.256\times128\,\mathrm{MiB}\times3)\rfloor=15$, exactly half. The S-66 point declares $n=8$, which is inside both.

Hypotheses. float32 operator ($4$ B/element; solve_triangular has no bf16 CUDA path — [U], settled by N-19, book 03), which is a property of the shipped shape and not of the record's arm, and the difference is stated rather than assumed away: READ ceq/arm_smprime.py::operator @ 99777ab returns torch.complex(num.real/zb, num.imag/zb) unconditionally, so the record's operator is complex at every switch setting including $g=0$, and RUN results/k_cert_local.json records "operator_dtype": "torch.complex64", "operator_bytes_per_element": 8 on every arm_smprime memory cell against "torch.float32" and 4 on every softmax cell. This book's shipped shape takes the $g\equiv0$ real branch of 08.2's Statement (if g == 0: return num.real / zb), so $4$ B/element is a specification of the shipped path, not a reading of the arm_smprime rows in this verse's own table — those rows are complex64 and are quoted as measurements of a different object (M-8). Every byte figure in this verse and in 08.20 that carries $4$ B/element is therefore conditional on that branch existing, and the branch is NOT MEASURED — needs one forward on the real path at $g=0$ (book 03, N-19); until it is read, the complex residency rule above is the conservative one. The sizing constants $C_{\rm RESIDUAL}=17.874$, $C_{\rm OPERATOR}=3.823$ at $R^2=0.996373$ were fitted on the signed arm and do not transfer to the shape (M-8; N-18 re-solves them).

Evidence. As tagged; Ruling 8 (Q2 dropped at $10.578$ GiB against $7.996$ GiB, READ V17K_RULINGS.md:328-339 @ 99777ab).

Mechanism. P-8; V-22; M-8; B15.

Kill. N-20/N-18 (book 03): measured over predicted outside $[0.9,1.1]$ for the shape's memory law at any $s\in[128,2048]$, or reserved above $7.996$ GiB at the declared $n$ for any S-66 point. Price: $\approx9$ GPU-min DERIVED (N-18) $+\approx3$ GPU-min (N-20). Planted negative: the complex base at $n=16384$ must read non-resident under the training loop, reproducing $10.578$ GiB within the allocator's spread.

If killed. Replacement 1 — the recompute backward (N-12): off-diagonal tiles rebuilt from $Q,K$ in the backward; the adjoint of the solve is one upper-triangular solve plus an outer product; retained memory linear in $s$ plus $z$. Kill: peak bytes still quadratic in $s$ — NOT MEASURED — needs the kernel. Replacement 2 — gradient checkpointing per block (the HF loop's gradient_checkpointing, measured $1287.5\to487.7$ MiB $=0.379\times$ at seq $1024$ on this card for the signed arm, READ ceq/hf/modeling_ceq.py::CEQModel.forward comment @ 99777ab; not transferable to the shape without N-18). Kill: the shape's ratio above $0.6$ at seq $1024$ — one measurement, $\approx1$ GPU-min DERIVED.

Terminal. The shape is quoted only at measured $(s,n)$ points; the resident $n$ per $s$ is the hard limit of every long-context card.

Part IX — Determinism and reproducibility#

08.22 — The identity manifest, the seed rule, and the bitwise forward regime#

Statement. Every cell the module produces carries the manifest of SHAPE §A.10 with these fields mandatory and hashed: beta, qk, g, gamma (init, final, Λ, verdict), diag_convention ∈ {kept, removed}, committor_route ∈ {dirichlet_gamma1, discounted_gamma}, route ∈ {solve_triangular, neumann_K, segmented, csr} (+K, +C), boundary_sets/goal_set/sink_set (sorted lists + sha256), K, m, t_star, S, D, d_model, n_train, n_eval, steps, seed, rng_plan, device, threads, dtype, torch_version, cublas_workspace, deterministic_regime, instrument_hash, manifest_hash, producer_cmd, delta_vec, delta_bare, delta_mask, one_over_1mg, V_inf, census, floor_exact, floor_zeroinfo, ceiling_hop_k, sky_depth, sky_width, sky_cot, chacal_gamma, chacal_diag, params_per_arm, param_residual_pct, void_contrasts, supersedes, plus the fields the two repair rounds make mandatory: weight_decay_groups ∈ {frozen_beta, grouped, ungrouped} (08.17; only frozen_beta is a cell — grouped leaves the gradient channel on $\beta_h$ open and halts at step $1$, ungrouped leaves both open, and both are refusals), rowsum_residual — the max over steps of $\|\mathrm{rowsum}(P)-1\|_\infty$ on the read layer's head $0$ with the step index of the max, training halting above $\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}$ at $s=64$ (08.3's third Kill row, not $\epsilon_{32}$, which is the conservation residual's bar and carries the amplification $|T|\cdot\|(I-Q)^{-1}\|_\infty=1{,}500$ this quantity does not) and the cell journalled halt = rowsum (08.17), beta_final — the final value of $\beta_h$ on the read layer's head $0$, journalled beside rowsum_residual so the decay channel and the gradient channel are separable and a row-sum reading names which one moved (08.17), eta_excursions_low and eta_excursions_high — two fields, because 08.5's projection covers only the lower boundary and a pooled counter is blind on the side that is not repaired (08.5), dz_degenerate_frac under 08.19 clause 1's restated predicate ($\gamma_h=0$ or $u_a^\top z=0$ for a clamp; $\Delta\tilde V\equiv0$ and $\Delta P\equiv0$ jointly for a rewrite — never $\Delta\tilde V\equiv0$ alone, which holds on $m/m$ clamps of every draw and drove this field to $1.000$ on the bed as registered) (08.16, 08.19), sink_indicator_convention ∈ {carries, zeroed} — mandatory because this book ships carries while 04_BEDS_AND_INSTRUMENTS.md:27 and S-10 register zeroed, and until the CORRECTIONS.md row exists every committor number of this book is NOT MEASURED on the bed as registered (08.3), and infeasible_frac, s_discard_frac (08.9, 08.10 — the two census lines whose frozen bar of $0.5$ decides bed admission), and the two loss guards 08.16 freezes, epsilon_dz ($\epsilon_\Delta=10^{-8}$, the $\mathcal L_\Delta$ denominator guard) and epsilon_q ($\epsilon_q=10^{-6}$, the $\mathcal L_c$ BCE input clamp) — journalled because the clamp changes the loss on exactly the draws where $\hat q$ attains $0$ or $1$ (R-07, R-24), which is where the gradient is largest, so a cell run at another value is a different cell. void_contrasts keeps S-13's full-matrix entry only beside its floor. The list is registered in docs/CEQ_SHAPE.md S-13 and journals never move (L-G2), so this book may not drop the entry; what it requires is that $\|\hat P-P_{\rm env}\|_\infty$ be journalled with the number $1.3853193786720772$ printed beside it as the floor the diagonal alone imposes (08.12), so no reader can take it for a measurement of learning, and that S-13's own kill K-D2 ("$\|\hat P-P_{\rm env}\|_\infty<10^{-3}$ on at least 6 of 8 seeds", READ docs/CEQ_SHAPE.md:1687-1691 @ 207e7b9) be recorded on the same cell as unreachable by $1{,}385\times$ rather than as unfired. The restatement on the off-diagonal sup $\max_{i\ne j}|\hat P_{ij}-P_{{\rm env},ij}|$ that 08.16's Replacement 1 uses is the repair, and it is a CORRECTIONS.md row owned by the coordinator: NOT WRITTEN — needs the coordinator's CORRECTIONS.md row against S-13 (08.12, CHARTER §7). floor_fano_k1 is struck from the mandatory set and may not be journalled as a floor at all: the tight Fano value $0.7124$ is admissible only where $I(s_0;a^\star)>0$, which is the negation of BED-S admission census line (9), so on every admitted draw it is not this bed's floor and a manifest carrying it as one is a refusal (08.9). A missing declared field is a refusal, not absent. Seed rule: one integer seed; model init from torch.manual_seed(seed); data generator seed+1, evaluation generator seed+2, drawn on the host and moved (ceq/lm.py::train_one, ceq/arm_smprime.py::bedm_draw); beds are generator + seed + expected sha256 regenerated in-notebook (Ruling 7); seeds $0$–$7$ deduplicated at $N=8$. Bitwise forward regime (Ruling 1): torch.use_deterministic_algorithms(True, warn_only=True), cudnn.deterministic=True, CUBLAS_WORKSPACE_CONFIG=:4096:8 exported before process start (set in-process it does not take, READ V17_R4_RETAKE_PRICE.md:178-181 @ 99777ab), allow_tf32 matmul False, one stream, threads=8; bitwise is required for replay and for every deciding forward cell; training inherits the backward hole and is compared against the measured noise floor. The cumprod clause this verse used to carry is deleted, because $g_h$ is not a trained parameter of the shipped module. It read "at the load corner ($g=0$) no cumprod is executed … reached only when $g\ne0$", which left the strict-mode hazard alive for every step after the first: had $g_h$ been learnable, one optimiser step off exactly $0.0$ would put the cumprod→cumsum backward — which raises under torch.use_deterministic_algorithms(True) (READ COSTS.md §1.6 @ 99777ab) — into every subsequent backward, and this Statement's own requirement of a bitwise deciding forward and Ruling 1's regime could not both hold (B13, Ruling 1, V-16). 08.13's Statement settles it the other way: $g_h\equiv0$ is frozen, non-learnable, so the hop is the all-ones causal mask by the blend at every step and not merely at load, no cumprod is ever executed in the shipped module, and the forward and its backward touch solve_triangular, matmul and softmax only. The strict-mode cumprod backward survives here as the planted negative of this verse's Kill and nowhere else. A cell journalled with $g\ne0$ is outside this book's shipped module; it must journal deterministic_regime = warn_only and it may not claim the bitwise bar, and no verse of this book is stated on it.

Hypotheses. One box, one torch version; solve_triangular was observed bitwise forward and backward over $8$ repeats under strict mode on this card with no torch-documented guarantee (RUN NEPTUNE, SHAPE §2.5) — an observation, not a contract.

Evidence. READ scale/identity_manifest.py:67-71,78,141-151 via SHAPE §A.10; the one-line-drift plant (flip any of beta/qk/g/gamma/boundary_sets by one unit → manifest_hash moves; READ V20_R15_WING_MANIFEST.md:148-152 @ 99777ab); the thread-count floor $2.345\times10^{-3}$ (READ scale/it11_verdict.py:133-137 via SHAPE §A.11 item 4); COSTS §1.6's three-row table (hop bitwise, forward bitwise, gradient NOT EXECUTABLE under the flag) [MEASURED].

Mechanism. B13; Ruling 1; M-10, M-16 (the lane is read from the journal, never the machine); P-1 (producer_cmd); V-16 (three outcomes — bitwise, drifting, raises — never "pass" on silence).

Kill. N-01 (book 03): $\max|\Delta|>0$ over $8$ repeats of the solve_triangular forward or backward under either flag regime on the certified device, or the backward raising under strict mode; frozen at $0.0$. Price: $\approx1$ GPU-s. Planted negative: the cumprod backward under strict mode must raise (it does, [MEASURED]), proving the instrument distinguishes "raises" from "bitwise".

If killed. Replacement 1 — warn_only for training, strict for every deciding forward: the shape's training cells are journalled deterministic_regime = warn_only like the scan arms, Ruling 1's bitwise bar is claimed for replay and forward only, and the noise floor is measured once (N-07, $\approx45$ s). Kill: the identical-seed NRMSE delta above $\mathrm{MDE}_8$ on any of $6$ pairs — the floor then swallows every kill in §5.5 and $N$ is repriced. Replacement 2 — replay by stored $P$: deciding cells store $P$ per head in float32 and replay the solve from the stored operator, so the forward is bitwise by construction regardless of the softmax kernel. Kill: stored-$P$ replay differing from the live forward by more than $0$ bits on one cell — a torch.equal, $0$ GPU-s.

Terminal. Bitwise forward on stored operators, training under warn_only with a printed floor; the canon licenses "replayable" and withdraws "the first gated wing whose training step runs strict".

Closing tables#

Census rows closed#

row	what is broken	verse(s) that close it	what the verse supplies
B26	no architecture document	08.1 (interface), 08.2–08.5 (operator, boundary rows, solve, dial), 08.6–08.11 (reads), 08.12 (traversal), 08.13–08.15 (block), 08.16–08.17 (training), 08.18 (data), 08.19 (API), 08.20–08.21 (budget), 08.22 (determinism)	the module on one page with shapes, constants, weights and prices, each decision with its kill and its pre-specified alternative
B13 (attachment)	`solve_triangular` bitwise but undocumented; `cumsum` raises	08.22	the regime; the no-`cumprod` corner at load; the N-01 instrument named (book 03) with its three-outcome kill
B14 (attachment)	unsynchronised timers, run order $\rho=+0.7029$	08.17, 08.20	every price tagged per-op floor; T1/T2 `NOT MEASURED` until S-66's two `synchronize()` calls and randomised order (book 03/05)
B15 (attachment)	the kernels do not exist; dense not runnable at $n=2048,s=4096$	08.20, 08.21	the FLOP/byte model with constants; $n$ per $s$; the chain fused → chunked (N-11) → dense per chunk (N-20) → cost sentence withdrawn (book 03)
B25 (attachment)	"faster to train" undefined and unmeasured	08.17, 08.20	T1 and T2 as the only licensed forms, at the LM shape only and at 08.15's Rule-B matched count $25{,}728{,}192$ vs $25{,}728{,}000$ ($192$ parameters, $0.00074627\%$, no re-architecting of any control), with the read layer's DERIVED ratio $1.517578\times$ the head at that geometry ($2.125\times$ being the $h=1$ design reading); their instrument (book 04) and cost law (book 03), their kill and the Terminal that withdraws both. The LM-shape step rate is `NOT MEASURED`, so the row is closed by a schedulable cell whose price is not yet readable, not by a Terminal

Kills, cheapest first#

The sort key, printed because the table used to be mis-sorted. Every price below is expressed in GPU-seconds including the arena or pair the kill depends on, and the dependency is printed in its own requires column. The order this table used to print was not cheapest-first and inverted three pairs: 08.11 at $\approx0.04$ GPU-min ($2.4$ s) stood above 08.5, whose $\le17.4$ s ablation runs on top of the $\approx98$ s arena ($\approx115.4$ s in total) and which in turn stood above 08.6's $\approx34$ s pair; 08.16's $\approx98$ s arena stood above 08.17's clause 1, which reads on the same arena; and 08.21's $\approx12$ GPU-min ($720$ s) stood above 08.20's $206$–$537$ GPU-s band, a factor of three at the widest. With 08.9's clause 1 repriced from $0$ GPU-s to the $\approx98$ s arena its comparator depends on (08.9's Kill), a fourth inversion appeared at row 4. All four are corrected here. Ten rows are $0$ GPU-s, and read against the requires column none of the ten needs the arena; the three rows whose price used to read as though it were free — 08.5, 08.9 clause 1 and 08.17 clause 1 — are priced at the arena they run on top of and are sorted there.

verse	kill number (frozen)	price (GPU-s, dependency included)	requires	replacement verse / route
08.15	clause 1: pair counts under the rule the cell journals differ by $>0.032\%$ (Ruling 3, R-15) — forfeits "at matched parameters" and every T1/T2 row; clause 2: $>0.1\%$ (S-30) — the pair may not be journalled at all. Rule B: LM $192/25{,}728{,}000=0.00074627\%$ matched, design $3/4{,}769=0.062906\%$ UNMATCHED $1.97\times$	$0$	the two parameter counts; no draw, no arm	delete the gate heads and $g_h$ from the module (Rule B and Rule L then coincide); kill is a different object: `torch.equal(O_with, O_without)` `False` on one draw at $g=0$, $0$ GPU-s. The MLP-widening replacement is deleted: at $8\times1025=8{,}200$ parameters it is $5.47\times$ the act Ruling 3 forbids by name
08.1	output tuple missing a field / shapes disagree with 08.1's table	$0$	one FOUND record of the arm's `kind`, plus the S-01 manifest field check	two-head form (H-q, H-z), kill stated per dtype
08.19	zero FOUND records with the arm's `kind` at end of Phase 2	$0$	the `results/` scan alone	08.1's two-head API
08.3	$\rho(Q)=1$ with the sink declared, or conservation residual above the dtype's bar: $\epsilon_{64}=10^{-8}$ float64, $\epsilon_{32}=1.5\times10^{-4}$ float32; and, as a third and distinct quantity, the operator row sum above $\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}$ at $s=64$ — never $\epsilon_{32}$, which carries the amplification $	T	\,\|(I-Q)^{-1}\|_\infty=1{,}500$ that a row sum does not, and which would admit a conservation residual of $3.5715\times10^{-3}$, $23.81\times$ its own float32 bar (DERIVED)	$0$
08.7	`route = neumann_K` on a committor cell; residual above 08.3's bar for the cell's dtype ($10^{-8}$ / $1.5\times10^{-4}$)	$0$	one declared draw	discounted read at $\gamma_{\rm env}$
08.18	leak probe $R^2\ge0.5$ honest, or $<0.99$ planted	$0$	the corpus and a CPU probe; no GPU	strip features → strike the task
08.9	clause 2 (census): R0 above the exact zero-information floor $0.875$ by more than its Clopper–Pearson interval	$0$	the untrained-arm 512-draw census; no arm, no arena	committor-vector head only → goal rule only
08.10	class frequency of $\phi$ outside $(0.05,0.95)$; $\mathrm{sd}(\mu)\le0.05$; $\hat\tau$ flat in $t^\star$; 08.9's `infeasible_frac` or `s_discard_frac` above $0.5$	$0$	the 512-draw census at $t^\star\in\{2,8,32\}$	$K=2$ isocommittor → hitting-time head
08.12	$t^\star=8$ unplaceable at $s=64$	$0$	the bed as registered	reroute the dial → `bed_1` as cross-check only
08.14	clause 1 restated on the committor, not on placement: fraction of admitted draws with $\max_{k\ge0}q^{(k)}_{i_q}>q^{({\rm sink})}_{i_q}$ below $0.216$ (`[ASSUMED]`, DERIVED from the record's own sink-share interval $[0.362,1.000]$; the placement form at $0.9$ is deleted — the geometry produces $0.34416$ at $p=3$ and $0.57987$ at $p=6$, so it fired on $100\%$ of censuses); clause 2: $\mathrm{sd}(q^{(k)}_{i_q})\le0.05$ on head $0$ of the read layer	$0$	the 512-draw census	boundary rows in every layer with the head axis restored → learned membership (leak-probed)
08.22	$\max\vert\Delta\vert>0$ or backward raises on `solve_triangular` over $8$ repeats	$\approx1$	the certified device under both flag regimes	`warn_only` + noise floor → stored-$P$ replay
08.2	`torch.equal` at the softmax corner `False`	$\le1.7$	one cell against the lane's own `softmaxAttn`	gate deleted, ChaCAL base (cited [V-eq], not [V])
08.4	solve vs dense $>10^{-12}$ float64; `torch.equal(O(0),PV)` `False` float32	$\le1.7$	one draw at $s=64$	Neumann with certificate → dense per chunk
08.13	loaded-corner logits not `torch.equal` at the LM shape	$\le1.7$	a pretrained softmax checkpoint to load	corner frozen → parallel branch with V-24's three substitutions
08.11	one exceedance on $1{,}024$ cells of $\delta_{\rm bare}+\delta_{\rm mask}$ with $\delta_{\rm mask}=\gamma\varepsilon/[(1-\gamma(1-\varepsilon))(1-\gamma)]$ (never the naive $\varepsilon/(1-\gamma)$, which under-reports by $4.74\times$); $\delta\Vert\tilde V\Vert_\infty\ge$ label sd	$\approx2.4$ ($0.04$ GPU-min)	$1{,}024$ drawn cells at $s=64$	exact route only
08.8	column route $\ge8\times$ one solve at $m=8$, synchronize-bracketed, median of $14$ after $2$ warm-ups	seconds, band only — the median is unpriced and the timer is book 03's	N-02's synchronised timer	cached mixture → per-row control
08.6	$r_{\rm softmax}/r_{\rm shape}\le2$ at marginal NRMSE within $\mathrm{MDE}_8$	$\approx34$	the $\approx34$ s arena pair (shape + control, $N=8$)	cost statement, kill at $1.12\times$ the measured $2.514$ ms $z$-channel total $=2.816$ ms at $n=2048$ (clock dispersion only; both model-derived candidates, $2.474$ ms total and $0.825$ ms increment, are printed in the verse and shown already crossed by the reference, so neither is frozen)
08.9	clause 1 (comparator): R0 flag rule $\ge\mathrm{acc}_{\rm shape}-\mathrm{MDE}_8$ on $6/8$	$\approx98$ — not $0$: R0 is oracle-free, but $\mathrm{acc}_{\rm shape}$ and $\mathrm{MDE}_8$ are the arena's	the $\approx98$ s eight-seed arena (08.16)	committor-vector head only → goal rule only
08.16	$\hat\gamma$ PINNED and $\mathcal L_q$ not moved by one seed sd	$\approx98$	the $\approx98$ s eight-seed arena	committor-only (kill on the off-diagonal sup and the induced $\varphi$-NRMSE, never the full-matrix sup, which fires by construction at $138.5\times$ the bar) → interventional pairs only
08.17	clause 1: `rowsum_residual` $>\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}$ at any arena step, read beside `beta_final` — the inference is "the operator is not row-stochastic", and `beta_final` names whether the decay channel or the gradient channel moved it; the old inference ("the optimiser is not running the parameter groups") could not distinguish the defect from the Statement working as specified	$\approx98$ — $0$ GPU-s of instrument on top of the arena it reads	the $\approx98$ s eight-seed arena	the frozen set is the Statement ($\beta_h,\kappa_h,g_h$, the two gate heads, `requires_grad=False`) → T2 only → per-step cost law
08.5	$\Lambda\le2.7055$ on $6/8$ seeds, admissible only on seeds where $\eta_h$ never left $[0,1]$ at either end, with `eta_excursions_low` and `eta_excursions_high` printed as two fields (the projection is $\mathrm{clip}(\eta_h,0,1)$; a one-sided projection leaves the upper trap, which pins $\gamma_{\max}$ at conditioning $1024$ and a conservation residual $25.6\times\epsilon_{32}$)	$\approx115.4$ ($\approx98$ arena $+\le17.4$ ablation)	the $\approx98$ s arena and the LR ablation cell	fixed $\gamma_{\rm env}$ → annealed dial with committor supervision
08.20	K-9: exponent interval excludes $2$ toward $3$; ratio $>3$ at $s\le1024$	$206$–$537$ (band)	the synchronised, order-randomised $s$-sweep (S-66, book 03)	chunked solve → dense per chunk with $n$ per $s$
08.21	memory law outside $[0.9,1.1]$; reserved $>7.996$ GiB at the declared $n$ (GiB against a card in GiB: `arm_smprime` at $n=16{,}384$ reads $7.504$ GiB reserved and `resident: true`; $10.578$ GiB is the training-loop number and is not resident)	$\approx720$ ($\approx9$ GPU-min N-18 $+\approx3$ GPU-min N-20)	the memory sweep over $s\in[128,2048]$	recompute backward → block checkpointing
08.17	clause 2: T1 ratio $>1$ on $6/8$ (interval excluding $1$), at the LM shape only, on BED-S at $t^\star=8,K=2,m=8$, matched count $25{,}728{,}192$ vs $25{,}728{,}000$ ($192$ parameters, $0.00074627\%$, Rule B, no widened control), $N=8$, skyline $\lfloor\log_2t^\star\rfloor+2=5$ — unlicensed at $d_{\rm model}=16$ under both counting rules, $1.97\times$ and $24.2\times$ the bar (08.15)	`NOT MEASURED` — needs the LM-shape step rate; the $\approx26$ s and $\approx64.8$ s figures are $d_{\rm model}=16$ rates and are not carried (V-22)	the LM-shape step rate on the certified device (`scripts/k_cert.py`, book 05)	T2 only → per-step cost law (K-9, no optimiser in the chain)

Limits, collected once. The safest-move rule of 08.9 is a one-step filter over evaluated candidates and not a policy; Bellman optimality can fail in multichain constrained MDPs with several unsafe sets (misra-2023-safety-constrained-mdp [V], CHARTER §5 B10, owned by book 01), and no policy-level optimality is claimed — 08.9's Evidence states that this citation is "carried in Limits" and until this round it was not, the whole book containing exactly one occurrence of it (RUN grep -n "misra\|Bellman\|multichain" docs/canon/08_ARCHITECTURE.md returned the Evidence line alone), which is a claim about the book's own contents that the book falsified (P-1, B10). No CEQ arm has been trained; every loss weight in 08.16 and every schedule constant is [ASSUMED] and frozen for the first arena only; the float32 conservation bar $\epsilon_{32}=1.5\times10^{-4}$ of 08.3 carries one unmeasured constant ($c$, NOT MEASURED — needs one float32 identity cell, S-22) and $c=1$ is the derivation with $1.68$ of headroom; $\epsilon_{\rm sink}=10^{-3}$ (08.9), $\mu_{\rm dp}=0.1$ (08.10), $\Lambda_{\mathcal F}=10$ (08.16) and the $0.216$ reachability bar (08.14) are [ASSUMED] and are to be re-registered from realised sd before any cell — and $\epsilon_{\rm sink}$'s printed reason was false by an order of magnitude for one round ("one order below $\mu_{\rm dp}=0.1$"; one order below $0.1$ is $10^{-2}$), so the value is kept and the reason is now the $6.67\times$ clearance over 08.3's $\epsilon_{32}$; the frozen set of 08.17 and the row-sum guard it is decided by have no cell and no record — the ungrouped AdamW of ceq/lm.py:264 and ceq/hf/train.py:378 is what the tree actually holds, so weight_decay_groups reads ungrouped on every existing record and every one of them is refused as a CEQ cell by this book's own rule, and a grouped cell with $\beta_h$ still learnable is refused too, because the gradient channel alone halts the guard at step $1$ at $327\times$ its bar; Ruling 2's "$\beta$ learnable, logged per instance" is not met by any cell of this book and every cell journals beta = frozen; $\hat\tau$ has no registered label, no loss term and no licensed comparison against any bed $\tau$, being inflated by up to $3.253722$ by the arm's own diagonal (08.10, 08.12), so one of the three components of "next equilibrium phase" is a diagnostic; the sink-indicator convention this book ships (carries) contradicts the bed as registered in 04_BEDS_AND_INSTRUMENTS.md:27 and S-10 (zeroed), the record being self-inconsistent because S-10's own $[0.362,1.000]$ sink share cannot exist under zeroed, and the CORRECTIONS.md row that would settle it is NOT WRITTEN — needs the coordinator (CHARTER §7), so every committor number of this book is NOT MEASURED on the bed as registered; S-13's K-D2 is unreachable by $1{,}385\times$ and its void_contrasts learnability reading is a diagonal constant $\ge1.3853193786720772$, both awaiting the same coordinator row; T1 and T2 are unlicensed at the $d_{\rm model}=16$ arena under both counting rules and their LM-shape price is NOT MEASURED, so this book schedules no T1/T2 cell it can price; every price is a per-op floor under a $2.0\times$–$6.6\times$ dispatch gap on a certificate taken at ab5b485 on torch 2.5.1+cu121 that the installed stack (2.14.0) does not reproduce (B1, B2); the Kaggle certificate is empty; the boundary-null citation for $2.7055$ is owed; solve_triangular's bitwise reading is one box, one version, no documented guarantee; the chess and market instantiations in 08.18 are interface examples with no bed, no floor and no matched baseline until book 09 registers them; the per-head $\beta$ and $\gamma$ granularity of 08.13 differs from the record's per-layer $\beta$ and is stated as this book's specification, not as a measured object.

Attacks answered#

MARS's findings, repair round 1, batch 1 of 2. Every finding appears verbatim — verse, flaw, mechanism, number, replacement_survives, severity — beside the repair. Nothing is softened; a finding that reads more gently after repair than when filed is itself a defect (V20_R15_JOURNAL.md corrections-index rule @ 99777ab). Where a finding could not be repaired the verse stays, marked OPEN, with its Terminal sentence in force.

A-1 · verse 08.2 · severity `strike` · mechanism V-25, V-3, P-7 · `replacement_survives: true`#

Flaw, verbatim. "The module ships in regime S ($\beta=1$): $P$ is real, causal, row-stochastic" is false whenever the learnable gate $g_h\neq0$, because the verse's own $Z_i$ omits the phase factor that its numerator carries. Number, verbatim. READ ceq/arm_smprime.py::operator @ 99777ab: num, mod = numerator(...), zb = mod.sum(-1, keepdim=True) ** beta, return torch.complex(num.real/zb, num.imag/zb); numerator returns (gh * e, rh * e) where gh = path_product(gate(m, th)) carries $e^{i\vartheta}$ and rh = path_product(m) does not. Row sum at $\beta=1$ is $\sum_j m_{ij}e^{i\vartheta_{ij}}e^{qk}/\sum_j m_{ij}e^{qk}$, a complex number of modulus $\le1$, equal to $1$ only at $\vartheta\equiv0$. The record's own measurement of the switch is $\max|g{=}1 - g{=}0| = 0.673101$ (08.2 Evidence), so the departure is $O(1)$, not roundoff.

Repair. 08.2's Statement now ships regime S at the frozen pair $(\beta=1,\ g\equiv0)$ and says row-stochasticity holds "only there"; a new DERIVED block, "Why $\beta=1$ alone does not give a row-stochastic $P$", carries the finding's three lines and its $0.673101$. 08.2's Hypotheses name $\beta=1$ and $g\equiv0$. The five dependent verses each carry $g\equiv0$ in their own Hypotheses by name: 08.3 (identity rows, conservation), 08.4 (hull bound, $\Pi_\gamma$ row sums), 08.7 ($q\in[0,1]$, conservation), 08.11 (certificate hypothesis), 08.12 (absorbing-chain reading). 08.13's Statement freezes $g_h\equiv0$ non-learnable in the shipped module with three reasons printed, one of them Ruling 1's (cumprod to cumsum backward raises under strict mode). The unfrozen-$g$ route is not restated anywhere: it is NOT MEASURED — needs the complex row-sum instrument (book 03/04).

A-2 · verses 08.13 / 08.15 / 08.17 · severity `strike` · mechanism M-8, V-25, D-3 · `replacement_survives: false`#

Flaw, verbatim. The training spec's AdamW decay applies to the learnable $\beta_h$, and the decayed $\beta$ breaks row-stochasticity within the arena's own 150 steps, firing 08.3's conservation kill and voiding 08.4, 08.7 and 08.11's hypotheses. Number, verbatim. 08.17 specifies weight_decay 0.01 and lr 3e-4 with no parameter groups; READ ceq/lm.py:264 @ 99777ab (opt = torch.optim.AdamW(m.parameters(), lr=lr)) and READ ceq/hf/train.py:378 @ 99777ab (opt = torch.optim.AdamW(model.parameters(), lr=lr)) — neither excludes any parameter. Decoupled decay gives $\beta\leftarrow\beta(1-\mathrm{lr}\cdot\mathrm{wd})^S=(1-3\times10^{-6})^S$: $\beta=0.99955$ at $S=150$ (the 08.17 arena) and $\beta=0.9716$ at $S=9{,}600$ (SHAPE M-5.3's ladder). Row sums become $Z_i^{1-\beta}$; at $s=64$, $Z_i\sim64$ this is $1.0019$ and $1.125$, i.e. conservation residuals of $1.9\times10^{-3}$ and $1.25\times10^{-1}$ against 08.3's frozen $10^{-8}$ — $1.9\times10^{5}$ and $1.25\times10^{7}$ times the threshold. 08.11's own READ V16_ARM_SMPRIME.md:28-32 @ 99777ab records rows summing $1.312192$ to $10.293107$ at $\beta=0$.

Repair. 08.17's Statement is now an AdamW with two parameter groups, printed as pseudocode, excluding $\beta_h,\kappa_h,g_h,\eta_h$ and the two gate heads from weight decay, with the finding's arithmetic carried as a four-line DERIVED block in its own numbers (both READ lines, $0.9995501005601203$, $0.9716107252156855$, $1.0018728307306244$, $1.1253202644447826$, $1.9\times10^{5}$ and $1.25\times10^{7}$ against $\epsilon_{64}$, and $12.7\times$ and $833\times$ against the float32 bar 08.3 derives). The alternative — $\beta_h\equiv1$ frozen — is 08.13's Replacement 1, which now states in its own text that it discharges this hazard by construction. A per-step row-sum guard $\|\mathrm{rowsum}(P)-1\|_\infty$ is added with the frozen halt bar $\epsilon_{32}=1.5\times10^{-4}$, is 08.17's Kill clause 1 with its own planted negative (an ungrouped run must cross the bar within 150 steps at $s=64$), and is a mandatory manifest field of 08.22 (rowsum_residual, with weight_decay_groups beside it, ungrouped a refusal). 08.16's Hypotheses require one of the two routes in force. On replacement_survives: false: both of 08.17's replacements now inherit the grouped optimiser from the Statement, said in the If-killed rather than assumed; Replacement 2 is additionally re-derived so that no optimiser is in its chain at all — it is the forward-only K-9 $s$-sweep on the $(s,n)$ ladder with no seeds-to-floor and no arena, a different number on a different object by a different instrument, which no weight-decay defect can reach.

A-3 · verse 08.15 · severity `strike` · mechanism Ruling 3, M-8, D-1, R-SKY · `replacement_survives: false`#

Flaw, verbatim. "Matched under Ruling 3" is claimed on the LM-shape pair while the verse's own cited design-geometry pair differs by 0.776 per cent, 7.8 times the verse's own frozen kill, and the arena on which every capability number is read runs at that geometry. Number, verbatim. 08.15 Evidence: 4,806-4,769=37 at $d_{\rm model}=16$ (READ ceq/hf/modeling_ceq.py::CEQAttention docstring @ 99777ab, confirmed at ceq/hf/modeling_ceq.py:409-418). $37/4{,}769 = 0.00776 = 0.776\%$. The book's own block adds 38 at that geometry ($38/4{,}769=0.797\%$). 08.15's Kill freezes $0.1\%$; Ruling 3 freezes $0.032\%$ (docs/CEQ_SHAPE.md:1397, :2270). 08.17's arena is steps=150, n_train=2048, n_eval=4096, s=64, d_model=16, seeds 0-7 — the same $d_{\rm model}=16$. So the pair is unmatched by $7.8\times$ the book's bar and $24\times$ Ruling 3's before any cell runs.

Repair. 08.15's Statement prints a five-row table of the ratio at every geometry the verse licenses, with both bars in the header: $0.7758\%$ at $d_{\rm model}=16$ (UNMATCHED, $24.2\times$ Ruling 3 and $7.8\times$ S-30), $0.032649\%$ for this book's block at the LM shape, $0.031996\%$ for the record's HF pair, and $0.000777\%$ for the LM shape with the control's MLP widened by $1$ hidden unit per block. Route (a) is taken at $d_{\rm model}=16$ and route (b) at the LM shape, and the verse prints why the If-killed cannot save the arena: one hidden unit costs $2d_{\rm model}+1=33$ parameters against a bar of $1.53$, so the closing move's granularity exceeds the bar by $21.6\times$ and the best integer widening still leaves $0.0833\%$, $2.6\times$ Ruling 3. A1, A2, T1 and T2 are therefore unlicensed at $d_{\rm model}=16$ and every capability sentence read on that arena is struck; the arena survives as a diagnostic cell for the kills that are not capability claims. The restriction is carried by name in 08.6's Hypotheses, 08.16's Hypotheses and 08.17's Hypotheses and Kill (T1/T2 at the LM shape only, priced NOT MEASURED), and Preface item 6 states it. The finding's $38$ is the $g_h$-counted variant and is carried in 08.15's Hypotheses as the $8{,}464$ / $0.032898\%$ LM row; the shipped module freezes $g_h$ (08.13), so the counted addition is $37$ and both are printed. On replacement_survives: false: the If-killed's widening move is retained and is explicitly declared not to save the arena — the correction of the already-published pair is the strike, not the widening.

A-4 · verse 08.3 · severity `strike` · mechanism V-17 (a threshold imported out of its units), M-2 · `replacement_survives: false`#

Flaw, verbatim. The conservation kill's frozen threshold $10^{-8}$ is a float64 number applied to the float32 pipeline the book ships, so it fires on every honest draw and decides nothing. Number, verbatim. 08.4 Hypotheses: "the solve is torch.linalg.solve_triangular in float32 on CUDA". 08.3's Evidence reads conservation to $4.4\times10^{-16}$ — a float64 reading (docs/CEQ_SHAPE.md:1673, RUN[SATURN]). Float32 unit roundoff is $5.96\times10^{-8}$, already above $10^{-8}$ from one rounding. With the verse-cited $\|Q\|_\infty=0.958$ (08.7 Evidence), $\|(I-Q)^{-1}\|_\infty\le 1/(1-0.958)=23.8$, and forward substitution over $|T|=63$ rows gives a conservation residual of order $63\times23.8\times5.96\times10^{-8}\approx 9\times10^{-5}$ — four orders above the frozen bar.

Repair. 08.3's Kill is now a two-row table, one bar per dtype: $\epsilon_{64}=10^{-8}$ for the float64 identity cell (eight orders above the record's own $4.4\times10^{-16}$), $\epsilon_{32}=1.5\times10^{-4}$ for the shipped float32 forward, the second DERIVED in three lines from the finding's own quantities ($2^{-24}=5.960464\times10^{-8}$; $\|(I-Q)^{-1}\|_\infty\le23.81$ from $\|Q\|_\infty=0.958$; $|T|=63$ giving $c\cdot8.94\times10^{-5}$, so $\epsilon_{32}$ freezes $c=1.68$ of headroom, with $c$ marked [ASSUMED] and NOT MEASURED — needs one float32 identity cell on the certified device, S-22). The bar is declared to rescale with $|T|$ and $\|(I-Q)^{-1}\|_\infty$ and to be re-derived, never carried (V-22), and a third planted negative tests the bar itself ($|T|=1200$ must read above $\epsilon_{32}$ and be repriced, not failed). 08.7's Kill now reads 08.3's bar for the dtype the cell ran in and states that its former single $10^{-8}$ was a float64 number applied in float32 that fires on every honest draw. On replacement_survives: false: 08.3's Replacement 2 and 08.1's replacement were the finding's own casualties and are repaired at the same edit — each carries two bars, $10^{-12}$ in float64 and $\epsilon_{32}$ in float32, and 08.3's Replacement 2 prints why a single $10^{-12}$ applied to the shipped path is $5.96\times10^{4}$ below one rounding and can never be met.

A-5 · verse 08.1 · severity `strike` · mechanism V-11 inverted (a kill that fires at every real draw), V-17 · `replacement_survives: false`#

Flaw, verbatim. The layer's replacement kill is unmeetable in the shipped dtype: it demands the conservation row to $10^{-12}$, four orders below float32 unit roundoff. Number, verbatim. 08.1 If killed: "an (H-q, H-z) record whose $q$ does not satisfy the conservation row to $10^{-12}$ (08.7) — decided on one draw, $0$ GPU-s". Float32 unit roundoff $= 2^{-24} = 5.96\times10^{-8}$, so a single rounding of a value near $1$ exceeds $10^{-12}$ by $5.96\times10^{4}$. The replacement therefore dies on its own kill on every draw, in the dtype 08.4 declares.

Repair. 08.1's If-killed kill is restated with its dtype named: $10^{-12}$ in float64 on the identity cell, or $\epsilon_{32}=1.5\times10^{-4}$ in float32 on the shipped forward, the float32 bar being the same DERIVED quantity 08.3 freezes and restated rather than carried. The verse prints the finding's arithmetic ($2^{-24}=5.960464\times10^{-8}$; the factor $5.96\times10^{4}$) and says in its own text that without both dtypes printed the (H-q, H-z) form is not a replacement (V-9) and 08.1's chain has one link, not two.

A-6 · verses 08.3 / 08.4 · severity `strike` · mechanism V-3, V-23, B26 · `replacement_survives: false`#

Flaw, verbatim. The sink's indicator channel is specified as zero, which makes $R_{\rm sink}=0$ and $q^{(\rm sink)}\equiv0$, contradicting the sink share both verses cite and destroying 08.7's conservation identity. Number, verbatim. 08.3 Statement: "The sink carries value $0$ on every indicator channel and is not a goal member." 08.4 Statement: "the sink channel forced to $0$ at position $0$." With $\mathcal A_{\rm sink}=\{0\}$ the indicator $\mathbb 1_{\mathcal A_{\rm sink}}$ has support $\{0\}$ only, so zeroing it at position $0$ zeroes the channel identically. Then $\hat q^{(\rm sink)}_T=(I-Q)^{-1}R_{\rm sink}\mathbb 1 = 0$ and $\sum_\bullet q^{(\bullet)} = 1 - q^{(\rm sink)}$, which on the judge's own draw is as low as $0$ because the sink share reads $[0.362, 1.000]$ (docs/CEQ_SHAPE.md:1673, RUN[SATURN]). 08.7's frozen residual kill ($>10^{-8}$) then fires on 100 per cent of draws.

Repair. Both Statements separate the two column groups by name. 08.3: the $d$ value columns are zeroed at position $0$ so BOS contributes nothing to $O$; the $K+2$ indicator columns are the set memberships unmodified, $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$, and the verse prints the consequence of conflating them in the finding's own terms including the $[0.362,1.000]$ reading and the 100-per-cent firing. 08.4's Statement and its pseudocode carry the same separation with three comment lines (Vtilde[0, :d] = 0; Vtilde[:, d:] = masks.T UNMODIFIED; Vtilde[0, d+0] == 1), and $R_{\rm sink}=P_{T,0}$ is printed explicitly as the causal-softmax weights from each transient position onto BOS, strictly positive on the whole of $T$ because the mask admits $j=0$ from every $i$. 08.7's Hypotheses require $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$ so that $R_{\rm sink}\ne0$ and $q^{(\rm sink)}$ is not identically zero. On replacement_survives: false: 08.3's Replacement 1 (column-sink token) now states the separation verbatim and declares that a replacement zeroing the indicator inherits the defect and is not a replacement (V-9); Replacement 2 (separate committor head) carries $\mathbb 1_{\mathcal A_{\rm sink}}[0]=1$ in its own Hypotheses.

A-7 · verse 08.1 · severity `strike` · mechanism B26 (the gap the book exists to close), P-4 · `replacement_survives: false`#

Flaw, verbatim. The committor, phase, move and margin outputs carry no head axis while $P$, $\gamma$, $\beta$ and $\delta$ are per head, so at $h>1$ the layer emits eight different chains and the book never says which one, or how they are aggregated. Number, verbatim. 08.1 shape table, LM shape $h=8$: $P$ is $[B,8,512,512]$ and $\delta$ is $[8]$, but $q$ is $[B,K+2,512]$, $\phi$ is $[B]$, $\mu$ is $[B]$, $\hat\tau$ is $[B]$ and the move scores are $[B,m]$ — all rank-deficient by one axis. 08.19 repeats it: out.q float32[B, K+2, s], out.phase (phi: int64[B], ...), out.gamma float32[h]. 08.2 Statement: "Per head, ... the layer forms $P=W_{\beta,qk,g}$", so there are $h=8$ distinct $Q$, $R_k$ and hence $h$ distinct committor vectors. No verse defines the reduction (mean over heads, a designated read head, or a learned combination).

Repair. The second option of the required repair is taken in its designated-read-head form, as a Statement and not a convention. 08.1 carries a "Head convention" paragraph: $P,\gamma_h,\beta_h,\kappa_h,\delta$ are per head; $q,\phi,\mu,\hat\tau,\Delta z$ and the move columns are not, because head $0$ alone of a boundary layer receives the row overwrite and carries the committor head while heads $1..h-1$ run at $\mathcal A=\{0\}$ — one committor chain per boundary layer at $h=8$, no reduction defined and none needed — and it states the finding's own consequence, that without the rule 08.9's argmin, 08.10's phase and 08.19's API are unbuildable at the LM shape. 08.14 fixes the same rule with every consequence enumerated ($O$ still concatenates all $h$ heads, so the residual stream and 08.13's load rule are untouched) and rejects the mean-over-heads alternative on the ground that $\arg\min$ and $\arg\max$ do not commute with it. The $h$-chain alternative is not left open: it is 08.14's Replacement 1, which restores the head axis ($q\to[B,h,K+2,s]$, $\Delta z\to[B,h,m,s,d]$, $\phi,\mu,\hat\tau$ and the four move columns each gaining a leading $h$), declares the reduction as part of itself (computed per head, all $h$ returned, the caller choosing; a mean refused), and carries a two-clause kill including the $h$ chains' pairwise disagreement fraction on $a^\dagger$ below $0.05$ on the 512-draw census.

A-8 · verse 08.9 · severity `strike` · mechanism V-16 (three outcomes, never a silent number), V-12 · `replacement_survives: true`#

Flaw, verbatim. The Chebyshev-conditioned column divides by $1-q^{(\rm sink)}$, which the verse's own cited draw attains at exactly zero. Number, verbatim. 08.9 Statement: $a^{\ddagger}=\arg\min_a\frac{\max_{k\ge1}q^{(k)}(a)}{1-q^{(\rm sink)}(a)}$. 08.3 Evidence and docs/CEQ_SHAPE.md:1673: "the sink share reads $[0.362,1.000]$ on the judge's draw" (RUN[SATURN]) — the upper endpoint is $1.000$, so $1-q^{(\rm sink)}=0$ and $a^{\ddagger}$ is $0/0$ or $\pm\infty$. 08.10 carries the same division: "for $K=2$ with the sink share removed this is the isocommittor $q^{(0)}/(1-q^{(\rm sink)})=\tfrac12$ band". No guard, epsilon or raise is specified for either.

Repair. 08.9's Statement defines the Chebyshev admission set $\mathcal S=\{a:q^{(\rm sink)}(a)\le1-\epsilon_{\rm sink}\}$ with $\epsilon_{\rm sink}=10^{-3}$ frozen, [ASSUMED]-tagged with its reason (one order below 08.10's $\mu_{\rm dp}=0.1$, so a move discarded here had a conditioned margin already below the decision-point band) and its re-registration clause from the realised sink-share sd; $a^\ddagger$ is the argmin over $\mathcal S$. A move outside $\mathcal S$ is discarded and counted, exactly as ties within $10^{-9}$ are, and the discarded fraction is a printed census line that enters 08.10's kill at a frozen bar of $0.5$. The bar is in 08.9's Hypotheses. 08.10's conditioned margin is computed only on positions admitted by the same bar, with the $[0.362,1.000]$ reading printed there. 08.19 states the raise: s_discarded[b,a] is returned per move and a call in which every move is discarded raises rather than returning an argmin over an empty set.

A-9 · verse 08.9 · severity `strike` · mechanism P-4 (a link that does not exist), B26 · `replacement_survives: true`#

Flaw, verbatim. No behaviour is specified when the subject condition of $a^\dagger$ admits no move, and the degeneracy structure makes an empty feasible set reachable at every $K$. Number, verbatim. 08.9 Statement: $a^\dagger=\arg\min_a\max_{k\ge1}q^{(k)}(a)$ subject to $q^{(0)}(a)\ge\max_{k\ge1}q^{(k)}(a)$. With sink, goal and $K$ constraints exhausting $\mathcal A$, $q^{(0)}+q^{(\rm sink)}+\sum_{k\ge1}q^{(k)}=1$, so $\max_{k\ge1}q^{(k)}\ge(1-q^{(0)}-q^{(\rm sink)})/K$ and the constraint requires $q^{(0)}(a)\ge(1-q^{(\rm sink)}(a))/(K+1)$ — at $K=2$, $q^{(0)}\ge0.333(1-q^{(\rm sink)})$. On the judge's draw at query 20, the Chebyshev minimum is $0.000$ attained by $9$ of $20$ targets of which $7$ carry sink share above $0.5$ (08.9 Evidence, RUN[M], R-07), so the feasible set is a live emptiness case, not a corner. out.move in 08.19 is int64[B] with no sentinel.

Repair. 08.9's Statement carries the finding's derivation verbatim under the heading "The feasible set $\mathcal F$, and what happens when it is empty", and a frozen rule: when $\mathcal F=\emptyset$ the layer emits $a^\dagger=a^\star$ with infeasible=True, never a sentinel and never a silent argmin over the empty set. The infeasible fraction over the 512-draw census is a printed census line beside the rule-disagreement fraction, is a mandatory manifest field (infeasible_frac, 08.22), and enters 08.10's census kill at a frozen bar of $0.5$ with its reason printed: on such a bed $a^\dagger$ is $a^\star$ under another name and the two columns cannot disagree, so the bed is not admitted. 08.19's out.move_cols gains infeasible: bool[B]. 08.16's $\mathcal L_{a^\dagger}$ skips and counts those draws rather than training against the fallback label, and 08.18's oracle field infeasible_star carries the same rule on the label side.

A-10 · verse 08.9 · severity `strike` · mechanism V-25 (a theorem whose hypothesis no draw satisfies), L-FLOOR · `replacement_survives: true`#

Flaw, verbatim. The hypothesis that makes the tight Fano floor admissible is the negation of the bed's own admission census line, so the floor $0.7124$ is inadmissible on every admitted draw. Number, verbatim. 08.9 Hypotheses: "$I(X_{\le0};a^\star)>0$ computed by plug-in so the tight Fano floor is admissible"; 08.9 Evidence: "tight Fano $0.7124$ where $I>0$". The BED-S admission census requires the opposite: census line (9) is "$I(s_0;a^\star)=0$ by plug-in" (READ docs/CEQ_SHAPE.md:1686 @ the S-12 card, docs/CEQ_SHAPE.md:1100 names it the M-21/M-18 repair). A draw with $I>0$ is a leaking draw and is discarded; a draw that is admitted has $I=0$ and only the exact zero-information floor $1-1/m=0.875$ applies.

Repair. The tight Fano row $0.7124$ is deleted from 08.9's floor set. The Evidence states that $1-1/m=0.875$ at $m=8$ is the only floor this book prints for the move columns, that $0.7124$ is admissible only where $I>0$ — the negation of admission census line (9) — and that it may be printed only as a diagnostic on the unadmitted, discarded, leaking draws where $I>0$ is measured, never as this bed's floor; the weak Fano $0.6667$ remains struck. 08.9's Hypotheses carry the admission line with both READ pins (docs/CEQ_SHAPE.md:1686, :1100). 08.9's Kill was already written on $0.875$ and is unaffected, as the finding says. 08.22 strikes floor_fano_k1 from the mandatory manifest set and declares a manifest carrying it as a floor a refusal.

A-11 · verses 08.9 / 08.16 / 08.19 · severity `strike` · mechanism D-2, M-1, V-8 · `replacement_survives: true`#

Flaw, verbatim. The move loss supervises the constraint-risk ranking against the goal-argmax label, so the column the API actually ships ($a^\dagger$) is trained by nothing. Number, verbatim. 08.16: $\mathcal L_a=\mathrm{CE}(\mathrm{softmax}_a(-\hat s(a)/\tau_a),\,a^\star)$ with $\hat s(a)=\max_{k\ge1}\hat q^{(k)}_{i_q}(a)$; the argmax of $\mathrm{softmax}(-\hat s)$ is $\arg\min_a\max_{k\ge1}q^{(k)}$, i.e. the UNCONSTRAINED $a^\dagger$. 08.9 defines $a^\star=\arg\max_aq^{(0)}(a)$, a different column, and the constrained $a^\dagger$ carries a subject condition the loss never sees. 08.19 ships out.move int64[B] (a_dagger). 08.18's table lists a_star as the only move label. So the gradient pushes the risk ranking toward the goal ranking, and neither the subject condition nor the shipped column appears in any loss.

Repair. Both halves of the required repair are applied. $\mathcal L_a$'s score becomes $\hat s(a)=-\hat q^{(0)}_{i_q}(a)$, so score and label are the same object and the argmax of the softmax is $\arg\max_a\hat q^{(0)}(a)$. A second CE term $\mathcal L_{a^\dagger}$ enters the loss at $\lambda_{a^\dagger}=1$: score $\hat s^\dagger(a)=\max_{k\ge1}\hat q^{(k)}_{i_q}(a)+\Lambda_{\mathcal F}(1-\mathbb 1[a\in\hat{\mathcal F}])$ with $\Lambda_{\mathcal F}=10$ frozen [ASSUMED] and $\hat{\mathcal F}$ the arm's own feasible set, so the penalty is differentiable in the committors it ranks; label the constrained oracle argmin a_dagger_star, which 08.18's table adds as a required oracle field with its definition, its $\mathcal F^\star$ and its empty-set rule. The old form is quoted in 08.16 and deleted with the finding's own reason printed. Stage A holds $\lambda_{a^\dagger}=0$ and stage B releases it with $\lambda_a$. 08.9 carries a "Which column ships, and which loss trains it" paragraph and 08.19 carries the required one line: out.move returns $a^\dagger$ and $\mathcal L_{a^\dagger}$ trains it; $\mathcal L_a$ trains out.move_cols.a_star and no other column.

A-12 · verse 08.11 · severity `strike` · mechanism L-CERT, V-3, V-17 · `replacement_survives: true`#

Flaw, verbatim. The masked-route certificate term is not a bound: the stated additive $\varepsilon/(1-\gamma)$ is 4.74 times smaller than the standard perturbation bound at the verse's own numbers, so an F1-masked certificate under-reports. Number, verbatim. 08.11 Statement: "An F1 mask with dropped row mass $\varepsilon$ adds $\varepsilon/(1-\gamma)$ to $\delta$." At the verse's own cited setting $s=3,\gamma=0.9,\varepsilon=0.1$ (RUN[J]) this reads $1.0$. The standard bound for $\tilde P=P-E$, $E\ge0$ with row sums $\varepsilon$, is $\|(I-\gamma\tilde P)^{-1}-(I-\gamma P)^{-1}\|_\infty\le\frac{\gamma\varepsilon}{(1-\gamma(1-\varepsilon))(1-\gamma)}=\frac{0.09}{0.19\times0.1}=4.7368$. The verse's own RUN, $0.5263157894736843$, is exactly $\varepsilon/(1-\gamma(1-\varepsilon))=0.1/0.19$ — the ATTAINED amplification on that instance, not a bound, and the verse presents neither number as what it is.

Repair. 08.11 carries a titled block, "The F1 mask term, derived, with the attained value printed beside it and neither confused for the other": the bound $\delta_{\rm mask}=\gamma\varepsilon/[(1-\gamma(1-\varepsilon))(1-\gamma)]$ as a DERIVED block in the finding's three lines ($A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ with $A=I-\gamma\tilde P$, $B=I-\gamma P$; $\|E\|_\infty=\varepsilon$, $\|B^{-1}\|_\infty\le1/(1-\gamma)$, $\|A^{-1}\|_\infty\le1/(1-\gamma(1-\varepsilon))$; multiply), evaluated at the verse's own setting to $4.7368421052631575$; the record's $0.5263157894736843$ printed as RUN and named as the attained amplification on that one instance, $9.0\times$ below the bound; and the naive $\varepsilon/(1-\gamma)=1.0$ named as neither, $4.74\times$ below the bound, with the finding's own consequence printed — an F1-masked certificate written with it under-reports and this verse's Kill would fire on the mask route by construction. 08.11's Kill now reads $\delta_{\rm bare}+\delta_{\rm mask}$ on the mask route with the DERIVED expression named and the naive form forbidden, and gains a second planted negative: the same cell certified with $1.0$ must read an exceedance while the same cell certified with $4.7368421052631575$ must not. delta_mask is a mandatory manifest field (08.22).

A-13 · verse 08.12 · severity `strike` · mechanism D-2, V-25, V-11 · `replacement_survives: false`#

Flaw, verbatim. "The walk of the oracle and the walk of the arm live in the same class" is false at the diagonal, and the falsehood makes 08.16's replacement kill unreachable. Number, verbatim. BED-S is specified "with self-loops only on declared absorbing positions" (READ docs/CEQ_SHAPE.md:1679, S-11), so $P_{\rm env,ii}=0$ for every $i\in T$. 08.2's Statement gives the arm "$P_{ii}>0$ for $i\ge1$" and 08.3's Evidence measures $\rho(Q)=\max_TP_{ii}=0.6926596893360386$ (RUN[J]). Hence $\|\hat P-P_{\rm env}\|_\infty\ge 2\max_TP_{ii}=1.385$ at the record's reading. 08.16's Replacement 1 kill is "$\|\hat P-P_{\rm env}\|_\infty$ not below $10^{-2}$ on 6 of 8" — it fires on 8 of 8 by construction at $138\times$ the bar. 08.12's own Evidence concedes the point for bed_1 ("$\rho(Q)=0.9408612510154677$ and SymmSupport, so no causal $\hat P$ equals it") and the Statement asserts the opposite for BED-S.

Repair. 08.12's Statement says the two walks share the causal order, not the class — both descend in token order, both are lower-triangular, both absorb on the declared sets, and they differ on the diagonal by construction — and prints the named residual $\|\hat P-P_{\rm env}\|_\infty\ge2\max_TP_{ii}=1.3853193786720772$ as a lower bound attained on the diagonal alone and independent of every off-diagonal entry the arm might learn, with the S-11 pin and the RUN[J] reading. It states that no training run closes the gap, that the concession the Evidence already made for bed_1 is now stated for BED-S rather than asserted away, and that every kill anywhere in this book written on the full-matrix $\|\hat P-P_{\rm env}\|_\infty$ fires by construction and is not a kill (V-11). Its Hypotheses carry "shared causal order, not shared class". On replacement_survives: false: 08.16's Replacement 1 kill is re-derived on two objects the diagonal does not decide — (i) the off-diagonal sup $\max_{i\ne j}|\hat P_{ij}-P_{{\rm env},ij}|$ at $10^{-2}$, frozen against the off-diagonal's own scale $\max_{i\ne j}P_{{\rm env},ij}\le1$, and (ii) the committor it induces, $\varphi$-NRMSE of $\hat q$ against $q^\star$ on the admitted region, with a bar from the realised paired sd ($\mathrm{MDE}_8$) rather than the un-anchored $10^{-2}$ — with a planted negative that an arm matching $P_{\rm env}$ off the diagonal and carrying the arm's own diagonal must pass clause (i).

A-14 · verse 08.14 · severity `strike` · mechanism V-10 (a gate satisfied by construction), V-11 · `replacement_survives: true`#

Flaw, verbatim. The kill cannot fire on any conforming task, because 08.18 makes the very condition it tests an input contract. Number, verbatim. 08.14 Kill: "any draw with a declared set after $i_q$ in the read layer (discard) exceeding a discard fraction of $0.5$". 08.18's data-interface table makes i_q "yes; $i_q\in T$, every set $<i_q$" a REQUIRED field of every task record, and 08.14's own Statement says "the layer asserts $i_q\in T$". A conforming task therefore has discard fraction $0.0$ on 100 per cent of draws, $0.5$ below the bar, on every bed as registered. Only the planted negative (masks shifted by $+1$) can move it, and a plant is not a kill (V-15 requires a plant IN ADDITION to a reachable kill, not instead of one).

Repair. The discard-fraction test is deleted as a kill and 08.14 prints the finding's reasoning in its own terms, including V-15's "in addition to, never instead of". Both clauses the required repair names replace it, on the 512-draw census at $t^\star=8,K=2,m=8,s=64$, both $0$ GPU-s, both failable by a conforming task: (1) reachability of the boundary rows from the query — the fraction of admitted draws whose read layer sees at least one declared set within $t^\star$ positions of $i_q$ falls below $0.9$ frozen, which a task can fail while placing every set legally at $<i_q$, in which case $q_{i_q}$ is the sink share and $L_{\rm b}=\{L-1\}$ is decoration; (2) the read head carries a non-constant committor — $\mathrm{sd}(q^{(k)}_{i_q})\le0.05$ across the 512 draws for any $k$ on head $0$ of the read layer, frozen at the same bar S-12 line (5) freezes, which decides the head-$0$ designation the Statement fixes. One planted negative per clause: masks shifted so every set sits at distance $t^\star+1$ must drive clause 1's fraction to $0.0$, and a read layer whose $P$ is the uniform causal row must drive clause 2's sd to $0.0$ on all $K+2$ channels.

A-15 · verse 08.19 · severity `strike` · mechanism V-16, V-10 · `replacement_survives: true`#

Flaw, verbatim. The API's raise-on-zero-displacement rule fires by construction at the corner the module is required to load and to train stage A at. Number, verbatim. 08.19: "A call whose dz is identically $0$ on any sequence raises rather than returning an undefined cosine (V-16)", with return_moves=True the signature default. 08.8's identity is $\Delta z(a)=\gamma\,c_a\,u_a^\top z/(1-\gamma u_a^\top c_a)$, which is identically $0$ at $\gamma=0$ for every move. 08.13's load rule sets $\gamma_h=0$ bitwise at load and 08.16's stage A freezes $\gamma_h$ at $0$ for $S_0=\lfloor S/4\rfloor=37$ of the arena's 150 steps. So every forward call carrying moves raises for the whole of stage A and at every load-parity cell. 08.8's Evidence records the same effect for constant values: "$V\equiv\mathbb 1\Rightarrow\Delta z\le10^{-15}$".

Repair. 08.19's rule is replaced by three outcomes printed as such, under the heading "The three raises and the two flags, separated, because a raise on an analytically predicted zero fires by construction", with the finding's $S_0=37$-of-$150$ arithmetic carried. (1) A predicted zero — $\gamma_h=0$ or $\Delta\tilde V\equiv0$, both zeros the identity predicts before the solve runs — returns $\Delta z=0$ with dz_degenerate[b,a]=True and does not raise. (2) An unpredicted zero — $\gamma_h>0$, $\Delta\tilde V\not\equiv0$, non-zero predicted magnitude, numerically zero return — raises; this is the only outcome the old rule should ever have fired on. (3) An all-discarded Chebyshev call raises; a batch with $\mathcal F=\emptyset$ does not, because that outcome is defined. out.dz_degenerate bool[B, m] is added to the API block, and 08.19's Hypotheses require $\gamma_h$ and $\Delta\tilde V$ readable before the solve, which is what makes clause 1 analytic and not a post-hoc excuse. 08.16's Hypotheses refuse the cosine term on flagged calls, not the whole forward, count them in the manifest field dz_degenerate_frac (08.22), and print the finding's own reason: a raise there aborts every step of stage A and every corner cell, a gate satisfied by construction and not a guard.

A-16 · verse 08.21 · severity `strike` · mechanism V-17, V-22, P-8 · `replacement_survives: true`#

Flaw, verbatim. The memory numbers are read out of results/k_cert_local.json in GB and printed as GiB, and the resulting residency verdict contradicts the certificate's own resident field and book 03's reading of the same cells. Number, verbatim. RUN python -c "import json;d=json.load(open('results/k_cert_local.json'))": softmax $n=32768$ reserved $=5{,}270{,}142{,}976$ B $=4.908$ GiB $=5.270$ GB; arm_smprime $n=16384$ peak $=6{,}759{,}747{,}072$ B $=6.296$ GiB and reserved $=8{,}057{,}257{,}984$ B $=7.504$ GiB, with "resident": true. 08.21 prints "softmax $0.302/0.380$ GiB at $n=2048$ ... $4.329/5.270$ GiB at $n=32768$" and "the complex corner-3 base ... $6.760/8.057$ GiB at $n=16384$ (reserved above the card ...)". 03_KERNEL.md:131 prints the same cells correctly: "softmax resident to $n=32768$ at $4.908$ GiB reserved" and "reserved $8{,}057{,}257{,}984$ B at $n=16384$". Book 08 then compares its GB figure $8.057$ against the card's $7.996$ GiB and concludes non-residency, where $7.504$ GiB $<7.996$ GiB and the certificate says resident.

Repair. 08.21's Statement is rewritten as a five-row table carrying raw bytes and the $2^{30}$ conversion and the certificate's own resident field per row, with the card stated three ways ($8{,}585{,}216{,}000$ B $=7.996$ GiB $=8.585$ GB, RUN on memory.cells @ 99777ab). The corrected row is exactly the required one: softmax $0.281/0.354$ GiB at $n=2048$ to $4.032/4.908$ GiB at $n=32{,}768$; arm_smprime $0.814/0.988$ GiB at $n=2048$, $6.296/7.504$ GiB at $n=16{,}384$, $12.559/13.951$ GiB at $n=32{,}768$. The residency sentence is rewritten: the complex corner-3 base is resident at $n=16{,}384$ outside a training loop ($7.504<7.996$ GiB, resident: true), the old sentence is deleted with its unit defect named against book 03's identical reading, and what survives is the training-loop reading $10.578$ GiB $=1.323\times$ the card and not resident — Ruling 8's Q2 drop, a training-loop number never to be confused with the certificate's forward+backward number; at $n=32{,}768$ the base is not resident on either reading. The $n$-per-$s$ rule, the $0.90$ factor ($7.196$ GiB) and the $1.256$ ratio are unaffected and the verse says so. The verse's other unit bridge is cut at the same edit: $2048\times64\times20\times4=10{,}485{,}760$ B $=10.0$ MiB $=10.486$ MB is printed apart from SHAPE N-20's $+8$ MiB, not joined to it by an "i.e.".

Consequential edits this batch made outside the verses the findings name#

where	what changed	which finding forced it
Preface item 6	states that no A1/A2 sentence is written in this book and that A1/A2/T1/T2 are licensed at the LM shape only	A-3
Preface item 9	the "no verse cites 08.23 or 08.24" claim made true — 08.17's Hypotheses cited a non-existent 08.24 and now cite 08.22	A-2 (inside the 08.17 rewrite)
08.5	$\eta_h$ named as excluded from weight decay, with the clamp trap it otherwise falls back into	A-2
08.6 Hypotheses	the $d_{\rm model}=16$ arena declared not a matched pair; its Kill read as a diagnostic contrast, no A1/A2 licensed at any geometry it names	A-3
08.13 Replacement 1	states that freezing $\beta_h$ discharges 08.17's weight-decay hazard by construction	A-2
08.17 Terminal	prints both MAC figures ($+50\%$ on the $z$ channel, $2.125\times$ the head for the read layer's total) instead of the smaller one alone, closing a contradiction 08.6's If-killed already named	A-2
08.18 table	`a_dagger_star` and `infeasible_star` added as required oracle fields; `a_star`'s definition printed beside them	A-11, A-9
08.22 manifest	six fields made mandatory (`weight_decay_groups`, `rowsum_residual`, `eta_excursions`, `dz_degenerate_frac`, `infeasible_frac`, `s_discard_frac`), `delta_mask` and `param_residual_pct` added, `floor_fano_k1` struck	A-2, A-9, A-10, A-12, A-15
"Kills, cheapest first"	twelve rows restated: dtype-indexed bars (08.3, 08.7); 08.9 split into its two clauses at their two real prices; 08.10's two new census lines; 08.14's two new clauses; 08.15's two bars and the move that does not close at $d_{\rm model}=16$; 08.17's new clause 1 and its repriced, geometry-restricted clause 2; 08.11's mask bound; 08.16's off-diagonal restatement; 08.21's GiB verdict	A-4, A-3, A-9, A-8, A-14, A-2, A-12, A-13, A-16
"Limits, collected once"	the unmeasured $c$ of $\epsilon_{32}$; $\epsilon_{\rm sink}$, $\mu_{\rm dp}$, $\Lambda_{\mathcal F}$ as `[ASSUMED]`; the parameter groups having no record, so every existing record reads `ungrouped` and is refused as a CEQ cell by this book's own rule; T1/T2 unpriced at the only geometry at which they are licensed	A-2, A-3, A-4, A-8, A-11

Open after this batch#

No finding of batch 1 is left open, and none is answered by a rename. Each replacement_survives: false carries a replacement re-derived on a different number, a different instrument or a different object: 08.17's Replacement 2 on the forward-only K-9 $s$-sweep with no optimiser anywhere in its chain (A-2); 08.15's If-killed retained and declared not to save the arena, the loss carried by the strike instead (A-3); dtype-indexed bars in place of one threshold imported out of its units (A-4, A-5); the indicator/value column separation inherited verbatim by both of 08.3's replacements, with a zeroing replacement declared a V-9 defect (A-6); 08.14's Replacement 1 restoring the head axis and declaring its own reduction rather than leaving it open (A-7); and 08.16's Replacement 1 restated on the off-diagonal sup and the induced committor, the full-matrix form being unreachable at $138.5\times$ the bar (A-13). Batch 2's findings are appended below this section.

MARS's findings, repair round 1, batch 2 of 2. Same rule: every finding verbatim — verse, flaw, mechanism, number, replacement_survives, severity — beside the repair. Six of the sixteen were already closed by batch 1's rewrites of the same verses; each of those six is recorded here with the line that now stands quoted, never with the finding softened, because a finding closed by another batch is still a finding filed against this book.

B-1 · verse 08.17 · severity `strike` · mechanism M-8, P-8, V-17 · `replacement_survives: true`#

Flaw, verbatim. The Terminal prices the read layer at $+50\%$ causal MACs, contradicting 08.20's own table, which totals the read layer at $2.125\times$ the head. Number, verbatim. 08.17 Terminal: "at $+50\%$ causal MACs per read layer (DERIVED, per-op floor)". 08.20's table at $s=64,d=16,K=2,m=8$: head $65{,}536$; total read layer $139{,}264 = 2.125\times$ the head, i.e. $+112.5\%$. The $+50\%$ figure is the $z$-channel solve alone ($s^2d/2=32{,}768$ over $s^2d$), which is what docs/CEQ_SHAPE.md:2054 (N-09) states: "the MAC prediction is plus 50 per cent on causal MACs ($3s^2d/2$ against $s^2d$)". A read layer additionally carries $s^2(K+2)/2=8{,}192$ indicator channels, $\le8{,}192$ for the committor head and $ms^2/2+msd=24{,}576$ for the moves. Required repair, verbatim. 08.17's Terminal must read "at $2.125\times$ the head's causal MACs per read layer at $(s,d,K,m)=(64,16,2,8)$, of which $+50\%$ is the $z$-channel solve (docs/CEQ_SHAPE.md:2054, N-09) and the remainder the $K+2$ indicator channels, the committor head and the $m$ move columns". The same $+50\%$ in 08.6's If-killed is correct as written because it prices the $z$ channel only, and must say so.

Repair. Closed in batch 1 inside the 08.17 rewrite (A-2). The Terminal now reads "at $+50\%$ causal MACs on the $z$ channel and $2.125\times$ the head for the read layer's total at $(s,d,K,m)=(64,16,2,8)$ (DERIVED, per-op floor; 08.6's If-killed and 08.20's table print the two figures and this Terminal quotes both rather than the smaller one)". 08.6's If-killed carries the disambiguation the finding demands: "$+50\%$ causal MACs on the $z$ channel alone ($s^2d/2=32{,}768$ over the head's $s^2d=65{,}536$ … and not the read layer's total, which is $2.125\times$ the head". Verified this batch against 08.20's table row, which reads $139{,}264$ ($2.125\times$ the head).

B-2 · verse 08.5 · severity `strike` · mechanism V-9, V-10, B12 · `replacement_survives: true`#

Flaw, verbatim. The dial's parameterisation makes $\gamma_h=0$ an absorbing state of the optimiser, so the verse's kill can be satisfied by an optimiser artefact independent of the horizon claim. Number, verbatim. 08.5: $\gamma_h=\gamma_{\max}\cdot\mathrm{clamp}(\eta_h,0,1)$, $\eta_h$ initialised at exactly $0$. torch.clamp's backward is $\mathrm{grad}\cdot\mathbb 1[\mathrm{min}\le x\le\mathrm{max}]$, so once $\eta_h<0$ the gradient is exactly $0$ and $\gamma_h$ is frozen at $0$ for the remainder of training. 08.17's decay then shrinks any recovered positive $\eta_h$ by $(1-3\times10^{-6})$ per step. 08.5's Kill is "$\hat\gamma$ PINNED, $\Lambda\le2.7055$ on at least 6 of 8 seeds" — the exact reading a dead one-sided gradient produces, so a PINNED verdict does not distinguish "the dial buys nothing" from "the parameterisation killed the dial". The verse names the hazard and exports it ("owned by book 02") rather than repairing its own kill. Required repair, verbatim. Reparameterise as $\gamma_h=\gamma_{\max}\,\sigma(\eta_h)$ with $\eta_h$ initialised at $-\infty$-free finite value giving $\gamma_h$ within one float32 ulp of $0$ (or keep the clamp and add a projection $\eta_h\leftarrow\max(\eta_h,0)$ after each step, which restores a two-sided gradient at the boundary). Add to 08.5's Kill a discriminating clause: the PINNED verdict is admissible only on seeds where $\eta_h$ never left $[0,1]$, and the count of seeds where it did is printed. Exclude $\eta_h$ from weight decay (see the 08.17 finding).

Repair. Closed in batch 1 (A-2's optimiser groups plus the 08.5 rewrite), by the second of the two routes the finding offers. 08.5's Statement now carries the projection $\eta_h\leftarrow\max(\eta_h,0)$ after every optimiser step as part of the Statement, with the clamp-trap derivation printed and the $\sigma$ form rejected for one printed reason (it cannot give $\gamma_h=0$ bitwise, and the bitwise load corner of 08.13 is the book's own load rule). 08.5's Kill carries the discriminating clause verbatim in force — "admissible only on seeds where $\eta_h$ never left $[0,1]$", the count journalled as 08.22's eta_excursions, a PINNED verdict on an excursion seed reported as an optimiser artefact and not counted, the cell repriced rather than decided if fewer than $6$ seeds are excursion-free — and a second planted negative (projection removed, $\eta_h$ initialised at $-10^{-6}$, which must read PINNED on $8$ of $8$ with $8$ excursions, or the counter is blind). $\eta_h$ sits in 08.17's weight_decay = 0.0 group. The closing table's 08.5 row prints the clause.

B-3 · verse 08.16 · severity `strike` · mechanism P-1 (a number with no class), V-16, B26 · `replacement_survives: true`#

Flaw, verbatim. Two of the six loss terms cannot be evaluated as written: $\epsilon$ has no value and no evidence class, and the BCE has no clamp although the verse's own neighbours record committors at exactly $0$ and exactly $1$. Number, verbatim. 08.16: $\mathcal L_\Delta=\frac1m\sum_a(1-\cos(\cdot))+\frac1m\sum_a\frac{\|\Delta\hat z(a)-\Delta z^\star(a)\|_2^2}{\|\Delta z^\star(a)\|_2^2+\epsilon}$ — $\epsilon$ is never given a value, a support or a class, while $\lambda_z=\lambda_q=\lambda_\Delta=\lambda_c=\lambda_a=1$ and $\tau_a=0.1$ all carry [ASSUMED]. $\mathcal L_c=\frac1{mK}\sum_a\sum_{k\ge1}\mathrm{BCE}(\hat q^{(k)}_{i_q}(a),y_k(a))$ with $\hat q$ produced by an exact linear solve: 08.9's Evidence records "a clamp into $\mathcal A_0$ gives $q^{(0)}=1.0000$ exactly (RUN[M], R-24)" and "the Chebyshev minimum is $0.000$ attained by $9$ of $20$" (RUN[M], R-07), so $\log(0)=-\infty$ on those draws and the loss is non-finite. Required repair, verbatim. Give $\epsilon$ a value and a class in the Statement (e.g. $\epsilon=10^{-8}$ [ASSUMED: float32 denominator guard, one ulp above $2^{-24}$ times the smallest admitted $\|\Delta z^\star\|_2^2$]) and specify the BCE input clamp $\hat q\leftarrow\mathrm{clamp}(\hat q,\epsilon_q,1-\epsilon_q)$ with $\epsilon_q$ frozen and its class, or use binary_cross_entropy_with_logits on a logit head. Add both constants to 08.22's mandatory manifest fields so the clamp is journalled.

Repair. 08.16's Statement renames the bare $\epsilon$ to $\epsilon_\Delta=10^{-8}$ frozen with its [ASSUMED] reason and its re-registration clause, and writes $\mathcal L_c$ on $\mathrm{clamp}(\hat q,\epsilon_q,1-\epsilon_q)$ with $\epsilon_q=10^{-6}$ frozen. A new paragraph, "The two guards without which two of the seven terms cannot be evaluated at all", carries the finding's own numbers: $q^{(0)}=1.0000$ exactly (R-24) and the Chebyshev minimum $0.000$ attained by $9$ of $20$ (R-07) as the draws on which $\log(0)=-\infty$ entered the sum, and the analytic $\Delta z^\star\equiv0$ of stage A as the draws on which the denominator was exactly $0$. The clamp's value is DERIVED ($-\log\epsilon_q=13.8155$ caps a term; $10^{-6}$ is $16.8\times$ float32 unit roundoff $2^{-24}=5.960464\times10^{-8}$, so rounding cannot push an interior committor onto the clamp while an exact endpoint always is). The logit-head alternative the finding offers is rejected with the reason printed: the committor head is an exact solve returning probabilities, not logits, and a logit parameterisation between solve and loss breaks the Dirichlet consistency the next paragraph asserts. Both constants are now mandatory manifest fields epsilon_dz and epsilon_q in 08.22.

B-4 · verse Preface (items 9, 15) · severity `strike` · mechanism P-4 (claimed scaffolding that does not exist), B26 · `replacement_survives: true`#

Flaw, verbatim. The preface routes four census attachments to verses 08.23 and 08.24, which do not exist in the book, and the closing table routes the same four rows to different verses. Number, verbatim. Line 15: "attachment points for B13 (08.24), B14 (08.22), B15 (08.22, 08.23), B25 (08.18, 08.22)". The book's last verse is 08.22 (RUN grep -n '^### 08\.' 08_ARCHITECTURE.md returns 08.1 through 08.22, 22 verses). The closing table at line 524-527 routes B13 to 08.22, B14 to 08.17 and 08.20, B15 to 08.20 and 08.21, B25 to 08.17 and 08.20 — four rows, four different mappings from the preface, and only B14's 08.22 overlaps at all (and it is wrong: 08.22 is determinism, 08.17/08.20 carry the timers). Two further dangling references: line 285 "which removes the cumprod from the forward at load (08.24)" and line 361 "Ruling 1's regime (08.24)". Required repair, verbatim. Replace the preface's item-9 mapping with the closing table's (B13 -> 08.22; B14 -> 08.17, 08.20; B15 -> 08.20, 08.21; B25 -> 08.17, 08.20) and repoint the two (08.24) references at line 285 and line 361 to 08.22. The book is 556 lines, well inside the 300-900 bound, so the missing verses are a citation defect and not a truncation.

Repair. Closed in batch 1 (A-2's 08.17 rewrite plus the preface edit). Preface item 9 now prints exactly the closing table's mapping — "B13 (08.22), B14 (08.17, 08.20), B15 (08.20, 08.21), B25 (08.17, 08.20) — the same mapping the closing table prints, and the only one" — followed by the verse census as a RUN: "The book has exactly $22$ verses, 08.1 through 08.22 … there is no 08.23 and no 08.24, and no verse cites one." Both dangling references are repointed. Verified this batch: RUN grep -n "08\.23\|08\.24" docs/canon/08_ARCHITECTURE.md returns four lines — the preface census sentence, batch 1's consequential-edits row, and this row's two quotations of the finding — and no verse citation, which is the property the preface asserts.

B-5 · verses 08.7 / 08.10 · severity `repair` · mechanism L-EQ, P-10, P-11 · `replacement_survives: true`#

Flaw, verbatim. The book's two most load-bearing formulas are carried by [V-cat] and [U] citations with no [V-eq] and no DERIVED block, which L-EQ makes inadmissible. Number, verbatim. 08.7 Evidence: "kemeny-1960-finitemarkov [V-cat], grinstead-1997-probability Thm 11.6 [U] own $B=NR$" for $\hat q^{(k)}_T=(I-Q)^{-1}R_k\mathbb 1$ — the formula every read in Parts II and III is a functional of. 08.10 Evidence: "$N\mathbb 1$ as expected absorption time: kemeny-1960-finitemarkov [V-cat], grinstead-1997-probability Thm 11.5 [U]" for $\hat\tau$. CHARTER §2 Number rule: "A [V] citation is inadmissible for a load-bearing statement; load-bearing needs [V-eq] — the theorem, its hypotheses, one numeric instance — or a DERIVED block (L-EQ, P-10)." [V-cat] and [U] are weaker than [V], not stronger. The same defect sits in 08.2's If-killed, whose entire replacement operator rests on "fagnou-2024-chacal Eq. 5, CITED [V]". Required repair, verbatim. Upgrade all three to [V-eq]: state Kemeny-Snell's absorbing-chain theorem with its hypotheses (finite chain, every state reaching an absorbing state, $Q$ substochastic with $\rho(Q)<1$) and one numeric instance — BED-1's is already in the book's own Evidence ($A=[0]$, $B=[1]$, $|T|=9$, resolvent read vs bed["q"] $0.0$, residual $1.04\times10^{-17}$, READ MATHEMATICS.md:542-600 @ 99777ab) — or write the three-line DERIVED block $q = R\mathbb 1 + Qq \Rightarrow (I-Q)q = R\mathbb 1$. Same for $\hat\tau$ ($\tau = \mathbb 1 + Q\tau$) and for ChaCAL Eq. 5.

Repair. Closed in batch 1 (A-6 for 08.7's block, A-1's chain for 08.2's If-killed, and the 08.10 rewrite). All three now carry both the [V-eq] triple and the DERIVED alternative the finding offers, so neither is load-bearing alone: 08.7's block states the Kemeny–Snell theorem, its three hypotheses with $\rho(Q)=0.6926596893360386$ read both ways, BED-1's numeric instance, and the three-line derivation $q=R_k\mathbb 1+Qq\Rightarrow(I-Q)q=R_k\mathbb 1$; 08.10's does the same for $\tau=\mathbb 1+Q\tau$ on the identical factorisation; 08.2's If-killed carries ChaCAL Eq. 5 as statement, hypotheses (including the undecided diagonal convention, B22) and one numeric instance at $\gamma=0$ and $\gamma=0.6$. Both catalogue entries are named in place as "[V-cat] and [U] … named as such, never as the load bearer".

B-6 · verse 08.1 · severity `repair` · mechanism B26, P-4 · `replacement_survives: true`#

Flaw, verbatim. The layer's output $O$ has one rank in the shape table and another in the API and the block, so an engineer cannot tell whether the head axis is concatenated inside or outside the layer. Number, verbatim. 08.1 shape table: $O$ is $[B,1,64,16]$ at the design geometry and $[B,8,512,64]$ at the LM shape, i.e. $[B,h,s,d]$. 08.19: out.O float32[B, s, d_model], rank 3. 08.13's block: x = x + o_proj( CEQAttention( LN1(x), ... ).O ) # heads concatenated, [B,s,d_model], so CEQAttention.O is already $[B,s,d_{\rm model}]$ before o_proj. Three statements, two ranks. Required repair, verbatim. Fix one convention and print it in all three places. The block's is the buildable one: CEQAttention.O is $[B,s,d_{\rm model}]$ after concatenation and before o_proj; 08.1's table should read $[B,64,16]$ and $[B,512,512]$, with a separate row for the per-head pre-concatenation tensor $O^h$ at $[B,h,s,d]$ if the book wants to price it.

Repair. Closed in batch 1 (A-7), by exactly the route the finding names. 08.1's table now carries two rows — "per-head read, pre-concatenation $O^h$ … $[B,1,64,16]$ / $[B,8,512,64]$" and "layer read, post-concatenation $O$ … $[B,64,16]$ / $[B,512,512]$" — and a Rank convention paragraph fixes the convention and states that it is printed identically in 08.13 and 08.19, with $O^h$ tabulated so 08.20's budget can price it.

B-7 · verses 08.15 / 06_PREDICTIONS · severity `repair` · mechanism M-2, Ruling 3 · `replacement_survives: true`#

Flaw, verbatim. Book 08 freezes the matched-parameter bar at 0.1 per cent while book 06 requires 0.032 per cent for the same object, so a pair at 0.05 per cent is matched under one book and unmatched under the other. Number, verbatim. 08.15 Kill: "counts differ by more than $0.1\%$ after the vector head is attached". 06_PREDICTIONS.md:265: A2 requires "$d_{\rm model}$ reduced until the count is within $0.032\%$ of the shape's (Ruling 3)"; 06_PREDICTIONS.md:162: "Matched at the recounted parameter count (R-15, within $0.032\%$)". docs/CEQ_SHAPE.md:2270 (R-15) freezes $0.032\%$; docs/CEQ_SHAPE.md:1909-1913 (S-30) freezes $0.1\%$; 04_BEDS_AND_INSTRUMENTS.md:130 carries both. 08.15's own Evidence quotes Ruling 3 verbatim ("$0.032\%$ residual is MATCHED") and then freezes a bar $3.1\times$ looser. Required repair, verbatim. 08.15 must print both bars and say which governs which sentence: $0.032\%$ is the bar for writing "at matched parameters" in an A1/A2/T1/T2 row (Ruling 3, R-15, book 06); $0.1\%$ is S-30's admission bar for journalling the control pair at all. Its Kill should freeze the tighter one for the capability claim.

Repair. 08.15's Statement already printed both bars per geometry in its table (batch 1, A-3); this batch rewrites its Kill into two clauses at the two bars with the sentence each governs named — clause 1 at $0.032\%$ (Ruling 3, R-15, book 06) forfeiting "at matched parameters" and every A1/A2/T1/T2 row, clause 2 at $0.1\%$ (S-30) refusing the pair as a journalled control at all — and prints the defect it replaces: "a pair at $0.05\%$ passed this book's kill and failed 06_PREDICTIONS.md:265's requirement on the identical object". A second planted negative is added: a pair constructed at exactly $0.05\%$ must pass clause 2 and fail clause 1, or the two bars have been collapsed again. 08.15's If-killed is retightened from $0.1\%$ to $0.032\%$, since a replacement whose kill is looser than the verse's is not a replacement (V-9). The closing table's 08.15 row prints both clauses.

B-8 · verse 08.21 · severity `repair` · mechanism M-8, P-8 · `replacement_survives: true`#

Flaw, verbatim. The DERIVED byte increment and the cited N-20 increment are presented as the same number and differ by 25 per cent because they count different channel widths. Number, verbatim. 08.21: "DERIVED at $n=2048$: $2048\times64\times20\times4=10.5$ MB, i.e. $+8$ MiB per $2048$ sequences as SHAPE §5.3 N-20 states". $2048\times64\times20\times4 = 10{,}485{,}760$ B $= 10.0$ MiB $= 10.486$ MB, not $8$ MiB. N-20's $8$ MiB is $2048\times64\times16\times4 = 8{,}388{,}608$ B exactly, i.e. the $d=16$ value channels only, without the $K+2=4$ indicator channels the book's $z$ carries (READ docs/CEQ_SHAPE.md:2084-2090, N-20). Required repair, verbatim. Print both: $+8.0$ MiB per 2048 sequences for the $d$ channels (N-20's number, for the $z$ the record's arm would carry) and $+10.0$ MiB for this book's $d+K+2=20$-channel $z$, with the $2.0$ MiB delta named as the indicator channels. Delete the "i.e." that equates them.

Repair. 08.21's Statement now prints the two increments apart with the delta named: $2048\times64\times16\times4=8{,}388{,}608$ B $=+8.0$ MiB for the $d=16$ value channels, which is N-20's figure at docs/CEQ_SHAPE.md:2084-2090 and the increment the record's own arm would carry; $2048\times64\times20\times4=10{,}485{,}760$ B $=+10.0$ MiB $=10.486$ MB for this book's $d+K+2=20$-channel $z$; and the $+2.0$ MiB delta as the $K+2=4$ indicator channels of 08.4 and nothing else ($2048\times64\times4\times4=2{,}097{,}152$ B, DERIVED). The "i.e." is deleted and the sentence it stood in is quoted as the defect, with the $25$ per cent named.

B-9 · verses 08.20 / 08.21 · severity `repair` · mechanism M-8, V-22, B15 · `replacement_survives: true`#

Flaw, verbatim. The byte budget assumes a float32 operator while the record's own operator, which the book cites as its base, is complex64 at 8 bytes per element, and no verse specifies a real-valued path at $g=0$. Number, verbatim. READ ceq/arm_smprime.py::operator @ 99777ab returns torch.complex(num.real/zb, num.imag/zb) unconditionally — the tensor is complex at every switch setting including $g=0$. RUN results/k_cert_local.json: every arm_smprime memory cell records "operator_dtype": "torch.complex64", "operator_bytes_per_element": 8, against "torch.float32" and 4 for softmax. 08.20 budgets "$P$ at $4\,s^2$ B (float32; $16$ KiB at $s=64$, $64$ MiB at $s=4096$)" and 08.21's Hypotheses assert "float32 operator ($4$ B/element)". At complex64 the $s=4096$ residency rule $n\le\lfloor 7.196\,\mathrm{GiB}/(1.256\times64\,\mathrm{MiB}\times3)\rfloor\approx30$ halves to $n\le15$. Required repair, verbatim. Add one line to 08.2's Statement specifying that at $g=0$ the operator is built and stored real (a if g == 0 branch returning num.real / zb, which is what makes the $\gamma=0$ corner bitwise anyway), and one line to 08.21's Hypotheses recording that the record's arm_smprime path is complex64 and that the shipped shape is not that path. Print both residency rules: $n\le30$ real, $n\le15$ complex, at $s=4096$.

Repair. All three edits made. 08.2's Statement carries the branch — "At $g\equiv0$ the operator is built and stored real — the branch if g == 0: return num.real / zb — which is what makes the $\gamma=0$ corner bitwise anyway and which fixes the byte width at $4$ B/element for 08.20 and 08.21" (added in batch 1 under A-1, verified here). 08.21's Hypotheses now record the record's dtype from the finding's own RUN, confirmed this batch (RUN grep -o '"operator_dtype": "[^"]*"' results/k_cert_local.json | sort | uniq -c $\to$ five torch.complex64, five torch.float32; the same split on operator_bytes_per_element $8$/$4$), state that the arm_smprime rows in this verse's own residency table are measurements of a different object (M-8), and mark the real branch NOT MEASURED — needs one forward on the real path at $g=0$ (book 03, N-19), with the complex rule declared the conservative one until it is read. Both residency rules are printed at $s=4096$: $n\le30$ real ($64$ MiB per head per sequence) and $n\le15$ complex ($128$ MiB), with the S-66 point's $n=8$ inside both. 08.20's byte line is split the same way, $4\,s^2$ B real against $8\,s^2$ B complex, $16$/$32$ KiB at $s=64$ and $64$/$128$ MiB at $s=4096$.

B-10 · verse 08.9 · severity `repair` · mechanism M-8, P-8, D-3 · `replacement_survives: true`#

Flaw, verbatim. The kill is priced at 0 GPU-s but its comparator requires the trained arm, so the price is short by the full arena and the closing table's ordering is built on it. Number, verbatim. 08.9 Kill: "R-24 frozen: the closed-form flag rule R0 ... at or above $\mathrm{acc}_{\rm shape}-\mathrm{MDE}_8$ on at least $6$ of $8$ arena seeds"; Price: "$0$ GPU-s (R0 is oracle-free)". R0 is oracle-free, but $\mathrm{acc}_{\rm shape}$ and $\mathrm{MDE}_8$ come from the eight-seed arena that 08.16 prices at $\approx98$ s and 08.17 at $\approx26$ s for two arms. The closing table repeats "$0$ GPU-s" and places 08.9 fourth in a list ordered "cheapest first". Required repair, verbatim. Reprice 08.9's kill as "$0$ GPU-s for R0 itself, inside the $\approx98$ s arena for the comparison" and move it in the closing table to sit beside 08.16. The second clause of the kill ("above the exact zero-information floor by more than its Clopper-Pearson interval") is genuinely $0$ GPU-s only on the untrained-arm census and should be listed separately.

Repair. Closed in batch 1 for the verse (A-9's rewrite splits the Kill into "two clauses at two prices, because one of them needs the trained arm and the other does not", clause 1 priced "$0$ GPU-s for R0 itself, which is oracle-free, but … decided inside the $\approx98$ s arena of 08.16 and is priced there", clause 2 "genuinely $0$ GPU-s" on the untrained-arm 512-draw census); completed this batch in the closing table, where clause 1 now carries $\approx98$ GPU-s with requires: the $\approx98$ s eight-seed arena (08.16) and sits immediately before 08.16, while clause 2 stays in the $0$ GPU-s block with requires: the untrained-arm 512-draw census; no arm, no arena.

B-11 · verse Closing table "Kills, cheapest first" · severity `repair` · mechanism CHARTER §2 depth rule, D-3 · `replacement_survives: true`#

Flaw, verbatim. The table is not ordered cheapest first: three inversions, one of them a factor of three. Number, verbatim. Row order and prices as printed: 08.11 "$\approx0.04$ GPU-min" ($2.4$ s) precedes 08.5 "$0$ GPU-s on the arena $+\le17.4$ s" (which requires the $\approx98$ s arena, so $\approx115$ s) which precedes 08.6 "inside the $\approx34$ s pair"; 08.16 "inside the $\approx98$ s arena" precedes 08.17 "$\approx26$ s $+64.8$ s skyline" ($\approx91$ s); 08.21 "$\approx12$ GPU-min" ($720$ s) precedes 08.20 "$206$-$537$ GPU-s (band)". With 08.9 repriced (see the 08.9 finding) a fourth inversion appears at row 4. Required repair, verbatim. Re-sort with every price expressed in GPU-seconds including the arena a kill depends on, and print the dependency in a new column ("requires"), so a reader can see that six of the ten "$0$ GPU-s" rows are $0$ GPU-s only after the $\approx98$ s arena has run.

Repair. The table is re-sorted with every price in GPU-seconds including its dependency, and a fifth column requires is added to all $24$ rows. The new order runs $0$ (ten rows: 08.15, 08.1, 08.19, 08.3, 08.7, 08.18, 08.9 clause 2, 08.10, 08.12, 08.14) $\to\approx1$ (08.22) $\to\le1.7$ (08.2, 08.4, 08.13) $\to\approx2.4$ (08.11) $\to$ seconds-band (08.8) $\to\approx34$ (08.6) $\to\approx98$ (08.9 clause 1, 08.16, 08.17 clause 1) $\to\approx115.4$ (08.5) $\to206$–$537$ (08.20) $\to\approx720$ (08.21) $\to$ NOT MEASURED (08.17 clause 2). A new sort-key paragraph names all four inversions with the finding's own numbers. One number in the required repair is not reproduced, and the correction is printed rather than absorbed: read against the new requires column, none of the ten $0$ GPU-s rows needs the arena — they are decided on parameter counts, a manifest check, one identity draw, a CPU probe or the untrained-arm 512-draw census — and the three rows whose price used to read as free (08.5, 08.9 clause 1, 08.17 clause 1) are the ones now priced at the arena and sorted there. The count is three, not six, and it is stated in the table's own paragraph so a reader can check it against the column.

B-12 · verse 08.9 / Limits · severity `repair` · mechanism P-1, B10 · `replacement_survives: true`#

Flaw, verbatim. The verse states that a citation is carried in the Limits paragraph, and the Limits paragraph does not carry it. Number, verbatim. 08.9 Evidence: "misra-2023-safety-constrained-mdp [V] carried in Limits (Bellman optimality can fail in multichain constrained MDPs — the policy-level object, not claimed)". RUN grep -n "misra\|Bellman\|multichain" docs/canon/08_ARCHITECTURE.md returns exactly one line, 199 — the Evidence line itself. The "Limits, collected once" paragraph at line 556 lists eight limits and none of them is the Bellman hazard. Required repair, verbatim. Append to the Limits paragraph: "the safest-move rule of 08.9 is a one-step filter over evaluated candidates and not a policy; Bellman optimality can fail in multichain constrained MDPs with several unsafe sets (misra-2023-safety-constrained-mdp [V], CHARTER §5 B10, owned by book 01), and no policy-level optimality is claimed."

Repair. The sentence is appended verbatim as the first limit of "Limits, collected once", followed by the finding's own RUN as the record of what the book had claimed about itself: the grep returned the Evidence line alone, which is a claim about the book's contents that the book falsified (P-1, B10).

B-13 · verses 08.1 / 08.11 · severity `repair` · mechanism L-CERT, D-3, P-4 · `replacement_survives: true`#

Flaw, verbatim. The certificate row asserts a non-zero $\delta$ unconditionally while the shipped route has $\delta=0$, and the truncation order $K'$ that the formula depends on has no default and no registered support anywhere in the book. Number, verbatim. 08.1 shape table, last row: "certificate | $\delta=\gamma^{K'+1}/(1-\gamma)$ per head | $[h]$". 08.11 Statement: "on the exact route $\delta=0$ and the triple is still printed", and 08.4's shipped route is solve_triangular. $K'$ appears in 08.11's formulas and once in 08.22's manifest as "route ∈ {solve_triangular, neumann_K, ...} (+K, +C)" — no default, no support, no relation to the $K$ of the constraint sets, which shares the letter. CHARTER §6's own scope sentence writes the certificate as $\gamma^{K+1}/(1-\gamma)$, conflating the two. Required repair, verbatim. Change 08.1's certificate row to "$\delta=0$ on route = solve_triangular (shipped); $\delta=\gamma^{K'+1}/(1-\gamma)$ on route = neumann_K with $K'$ the journalled truncation order", register $K'$'s support in 08.11's Hypotheses (the record's read is $K'\in\{1,2,4,8,16\}$, 08.11 Evidence RUN[M]), and add one line noting that this book's $K+2$ value channels and $K'$ Neumann order are two objects the CHARTER §6 scope sentence writes with one letter.

Repair. Closed in batch 1 (A-12), by all three of the finding's clauses. 08.1's table row reads "certificate | $\delta$ per head, route-dependent", with a Certificate row, in full paragraph beneath it: "$\delta=0$ on route = solve_triangular, the shipped route (08.4); $\delta=\gamma^{K'+1}/(1-\gamma)$ on route = neumann_K, where $K'$ is the journalled truncation order with registered support $K'\in\{1,2,4,8,16\}$ (08.11). $K'$ (the Neumann order) and $K$ (the number of constraint sets, hence $K+2$ value channels) are two objects; CHARTER §6's scope sentence writes both with one letter, and this book does not." 08.11's Hypotheses register the same support from the record's own RUN[M] read and name the per-cell journal field.

B-14 · verses 08.13 / 08.22 · severity `repair` · mechanism B13, Ruling 1, V-16 · `replacement_survives: true`#

Flaw, verbatim. The gate switch is specified as a trained parameter and as an object whose backward raises under the determinism regime the book requires for every deciding forward, with no verse freezing it. Number, verbatim. 08.15 counts $g_h$ among the $3h$ added trainable switches; 08.13's Replacement 1 ("the corner frozen: $g_h\equiv0$ ... non-learnable") implies the Statement's $g_h$ is learnable. 08.22: "At the load corner ($g=0$) ... no cumprod is executed ... the cumprod→cumsum backward, which raises under strict mode (READ COSTS.md §1.6 @ 99777ab), is reached only when $g\neq0$." Once one optimiser step moves $g$ off exactly $0.0$, every subsequent backward under torch.use_deterministic_algorithms(True) raises. 08.22's Statement requires "bitwise ... for replay and for every deciding forward cell". Required repair, verbatim. State in 08.13's Statement whether $g_h$ is frozen in the shipped module. If frozen (the reading the rest of the book needs — see the 08.2 row-stochasticity finding), drop $g_h$ from 08.15's added-parameter count ($38\to37$ per block at $d_{\rm model}=16$, $8{,}464\to8{,}456$ at the LM shape) and delete the cumprod clause from 08.22. If learnable, 08.22 must journal deterministic_regime = warn_only for every training cell and 08.2 must ship the complex row-sum guard.

Repair. The frozen branch is taken, and all three of its consequences are now in the book. 08.13's Statement freezes $g_h\equiv0$ non-learnable in the shipped module — "not '$0$ at load'" — with three printed reasons, the third being this finding's (batch 1, A-2/A-1). 08.15 drops $g_h$ from the count: $37$ per block at $d_{\rm model}=16$ ($34+1+2$) and $8{,}400$ at the LM shape ($8\times(1026+8+16)$), with the counterfactual retained in its Hypotheses ("or the counts above are each one per head per block higher and the LM-shape row moves to $8{,}464$"). This batch deletes 08.22's cumprod clause and prints the deletion: the clause left the strict-mode hazard alive for every step after the first, 08.13 settles it the other way, no cumprod is ever executed in the shipped module, the strict-mode cumprod backward survives only as this verse's planted negative, and a cell journalled with $g\ne0$ is outside the shipped module, must journal deterministic_regime = warn_only and may not claim the bitwise bar.

B-15 · verse 08.6 · severity `repair` · mechanism P-1, M-2 · `replacement_survives: true`#

Flaw, verbatim. The replacement's kill threshold is a bare multiplier with no derivation and no evidence class. Number, verbatim. 08.6 If killed: "Kill: the increment at $n=2048$ above $2.2\times$ the reference $1.041$ ms (N-02)". The reference $1.041$ ms carries RUN NEPTUNE ($2.514-1.473$, 08.20). The multiplier $2.2$ carries nothing — no DERIVED arithmetic, no [ASSUMED] reason, no RUN. N-02's own plant tolerance in 08.20 is $\pm12\%$ and the dispatch gap in the same verse's Hypotheses is $2.0\times$-$6.6\times$, so $2.2$ sits inside the gap band and cannot separate a real regression from dispatch. Required repair, verbatim. Either derive the multiplier from the MAC model (the read layer's $z$-channel prediction is $1.5\times$ the head, so a bar at $1.5\times(1+0.12)=1.68\times$ separates the model from the clock) or tag it [ASSUMED: <reason>], and state which end of the $2.0\times$-$6.6\times$ dispatch band the bar sits inside so a reader knows the kill cannot fire on dispatch alone.

Repair. Closed in batch 1 inside the 08.6 rewrite (A-3's chain), by the first of the two routes. 08.6's If-killed reads "with the multiplier derived rather than asserted: the increment at $n=2048$ above $1.68\times$ the reference $1.041$ ms (N-02). DERIVED: the $z$-channel MAC prediction is $1.5\times$ the head, and the clock is non-stationary at $\pm12\%$ (08.20 Hypotheses), so $1.5\times1.12=1.68\times$ is the tightest bar that the model plus the clock cannot cross on their own. The bar sits at the bottom of the $2.0\times$–$6.6\times$ dispatch band …, $0.32\times$ below its lower edge, so a crossing is a real regression and not dispatch — which is exactly what a bare $2.2\times$ could not say, sitting inside the band with no derivation, no [ASSUMED] reason and no RUN (P-1, M-2)." The closing table's 08.6 row carries the $1.68\times$ bar.

B-16 · verse Preface (item 6, line 12) · severity `repair` · mechanism P-1, P-7 · `replacement_survives: true`#

Flaw, verbatim. The preface makes a false statement about the book's own contents: it names three verses as the only homes of the two licensed phrases, and one of them is the only home while the other two contain neither. Number, verbatim. Line 12: "'More accurate' and 'faster to train' appear only in the licensed forms A1/A2/T1/T2 of CHARTER §1, and only in verses 08.17, 08.18 and 08.22." RUN grep -nE "more accurate|faster to train|beats|edge|outperform" docs/canon/08_ARCHITECTURE.md: five hits at lines 12 (this preface item), 367 and 372 (both inside 08.17), 395 (08.18, but the hit is the word "faster" appearing nowhere — line 395's match is on "edge tokens", a different word), and 527 (the closing B25 row). "More accurate" appears at line 12 and nowhere else in the book. 08.22 contains neither phrase. Required repair, verbatim. Rewrite item 6 as: "'faster to train' appears only in 08.17 (Kill and Terminal) and in the closing B25 row, always as T1/T2 with the floor, $N=8$ and the depth skyline $\lfloor\log_2t^\star\rfloor+2=5$ named; 'more accurate' appears in no verse, and no A1 or A2 sentence is written in this book." Also fix 08.17's Kill, which names T1/T2, the floor and $N$ but neither the bed nor the matched count, both of which CHARTER §1 requires in the same row.

Repair. The first half was closed in batch 1 (A-3): item 6 now reads "'Faster to train' appears only in 08.17 (Kill and Terminal) and in the closing B25 row, always as T1/T2 with the floor, $N=8$, the bed, the matched count and the depth skyline $\lfloor\log_2t^\star\rfloor+2=5$ named in the same row … 'More accurate' appears in no verse of this book, and no A1 or A2 sentence is written here". The second half is closed this batch: 08.17's Kill clause 2 now names the bed (BED-S as registered at $t^\star=8$, $K=2$, $m=8$; SHAPE §5.3 S-11, S-62) and the matched count ($25{,}736{,}400$ shape against $25{,}736{,}200$ widened control, residual $200$ parameters $=0.000777\%$, $41\times$ inside Ruling 3's $0.032\%$) beside the floor, $N=8$ and the skyline $5$, with the CHARTER §1 requirement it was failing printed in the clause. The closing table's 08.17 clause-2 row carries the same five fields, so the B25 row and the verse now agree field for field.

Open after this batch#

No finding of batch 2 is left open. Six were already closed by batch 1's rewrites of the same verses (B-1, B-2, B-4, B-5, B-6, B-13) and are recorded above with the line that now stands, not with the finding softened; two more were half-closed and are completed here (B-10's closing-table half, B-16's 08.17-Kill half); the remaining eight are repaired in this batch. Every replacement_survives in this batch reads true, so no replacement had to be re-derived on a new object; the one place a replacement was nonetheless retightened is 08.15's If-killed, moved from $0.1\%$ to $0.032\%$ because a replacement whose kill is looser than its verse's is a V-9 defect in itself. One number in a required repair is not reproduced and the disagreement is printed rather than absorbed: B-11 asks the table to show that six of the ten $0$ GPU-s rows need the arena, and the requires column shows that none of the ten does, three other rows being the ones that do; the count is stated in the table so a reader can check it column by column.

tensor	symbol	\(s=64,d=16,h=1,K=2,m=8\)	LM shape (\(s=512\), \(d_{\rm model}=512\), \(h=8\), \(d=64\))	S-66 long point (\(s=4096\), \(d=16\), \(h=1\), \(n=8\))
queries, keys, values	\(Q,K,V\)	\([B,1,64,16]\) each	\([B,8,512,64]\)	\([B,1,4096,16]\)
gate heads	\(u,\theta\)	\([B,64]\) each	\([B,512]\)	\([B,4096]\)
operator	\(P\)	\([B,1,64,64]\)	\([B,8,512,512]\)	\([B,1,4096,4096]\)
set masks	\(\mathcal M\)	\([B,4,64]\)	\([B,K+2,512]\)	\([B,K+2,4096]\)
stacked right-hand side	\(\tilde V=[V\,\\|\,\mathbb 1_{\mathcal A_\bullet}]\)	\([B,1,64,20]\)	\([B,8,512,64+K+2]\)	\([B,1,4096,16+K+2]\)
state	\(z\)	\([B,1,64,20]\)	\([B,8,512,64+K+2]\)	\([B,1,4096,16+K+2]\)
per-head read, pre-concatenation	\(O^h\)	\([B,1,64,16]\)	\([B,8,512,64]\)	\([B,1,4096,16]\)
layer read, post-concatenation	\(O\)	\([B,64,16]\)	\([B,512,512]\)	\([B,4096,16]\)
committor vector (head \(0\))	\(q\)	\([B,4,64]\)	\([B,K+2,512]\)	\([B,K+2,4096]\)
displacement per move (head \(0\))	\(\Delta z\)	\([B,8,64,16]\)	\([B,m,512,64]\)	\([B,m,4096,16]\)
move scores (head \(0\))	\(\max_{k\ge1}q^{(k)}(\mathrm{do}\,a)\)	\([B,8]\)	\([B,m]\)	\([B,m]\)
phase (head \(0\))	\((\arg\max_k q_{i_q}^{(k)},\ \text{margin},\ \hat\tau)\)	\([B]\), \([B]\), \([B]\)	same	same
certificate	\(\delta\) per head, route-dependent	\([h]\)	\([8]\)	\([1]\)

geometry	rule	control	shape	added	residual	vs Ruling 3 (\(0.032\%\))	vs S-30 (\(0.1\%\))
design, \(d_{\rm model}=16\), \(h=1\)	B (shipped)	\(4{,}769\)	\(4{,}772\)	\(3h=3\)	\(3/4{,}769=0.062906\%\)	UNMATCHED, \(1.97\times\)	matched
design, \(d_{\rm model}=16\), \(h=1\)	L	\(4{,}769\)	\(4{,}806\)	\(2(d_{\rm model}{+}1)+3=37\)	\(37/4{,}769=0.775844\%\)	UNMATCHED, \(24.2\times\)	UNMATCHED, \(7.8\times\)
design, \(d_{\rm model}=16\), the record's arm	L	\(4{,}769\)	\(4{,}806\)	\(37\)	\(0.775844\%\)	UNMATCHED, \(24.2\times\)	UNMATCHED, \(7.8\times\)
LM shape, \(d_{\rm model}=512\), \(h=8\), \(L=8\)	B (shipped)	\(25{,}728{,}000\)	\(25{,}728{,}192\)	\(8\times3h=8\times24=192\)	\(192/25{,}728{,}000=0.00074627\%\)	matched, \(42.9\times\) inside	matched
LM shape, per-layer switch granularity	B	\(25{,}728{,}000\)	\(25{,}728{,}024\)	\(8\times3=24\)	\(24/25{,}728{,}000=0.00009328\%\)	matched, \(343\times\) inside	matched
LM shape, this book's block	L	\(25{,}728{,}000\)	\(25{,}736{,}400\)	\(8\times(1026{+}8{+}16)=8{,}400\)	\(8{,}400/25{,}728{,}000=0.032649\%\)	UNMATCHED, \(1.02\times\)	matched
LM shape, per-layer switch granularity	L	\(25{,}728{,}000\)	\(25{,}736{,}232\)	\(8\times1029=8{,}232\)	\(8{,}232/25{,}728{,}000=0.031996\%\)	matched, inside by \(1.00012\times\)	matched
LM shape, the record's HF pair	L	\(25{,}728{,}000\)	\(25{,}736{,}232\)	\(8{,}232\)	\(0.031996\%\)	matched, inside by \(1.00012\times\)	matched

stage	MACs	at \(s=64,d=16\)	at LM \(s=512,d=64\)	depth
softmax head (\(QK^\top\) causal \(+\,PV\) causal)	\(s^2d\)	\(65{,}536\)	\(16{,}777{,}216\)	1
gate path product (only when \(g\ne0\); never in the shipped module)	\(s^2/2\)	\(2{,}048\)	\(131{,}072\)	\(s\) (cumprod)
the solve, \(d\) value channels	\(s^2d/2\)	\(32{,}768\)	\(8{,}388{,}608\)	\(s\)
per-head subtotal (shipped, \(g\equiv0\))	\(\tfrac32s^2d\)	\(98{,}304\)	\(25{,}165{,}824\)	\(s\)

stage	MACs	at \(s=64,d=16,K=2,m=8\)	at LM \(s=512,d=64,K=2,m=8\)	depth
the solve, \(K+2\) indicator channels	\(s^2(K+2)/2\)	\(8{,}192\)	\(524{,}288\)	\(s\)
committor head at \(\gamma=1\) on \(T\)	$	T	^2(K+2)/2\le s^2(K+2)/2$	\(\le8{,}192\)
\(m\) clamp moves (column route)	\(m\,s^2/2+m\,s\,d\)	\(16{,}384+8{,}192\)	\(1{,}048{,}576+262{,}144\)	\(s\)
head-\(0\) subtotal	\(s^2(K+2)+ms^2/2+msd\)	\(40{,}960\)	\(2{,}359{,}296\)	\(s\)

geometry	layer total	softmax layer	ratio
design, \((s,d,K,m,h)=(64,16,2,8,1)\)	\(98{,}304+40{,}960=139{,}264\)	\(65{,}536\)	\(\mathbf{2.125\times}\)
LM shape, \((s,d,K,m,h)=(512,64,2,8,8)\)	\(8\times25{,}165{,}824+2{,}359{,}296=203{,}685{,}888\)	\(8\times16{,}777{,}216=134{,}217{,}728\)	\(\mathbf{1.517578\times}\)

cell	dtype	frozen bar	how the bar is fixed
the identity cell (08.3, 08.7)	float64	\(\epsilon_{64}=10^{-8}\)	the record's own reading is \(4.4\times10^{-16}\) (Evidence), eight orders below the bar
the shipped forward (08.4's `solve_triangular` on CUDA)	float32	\(\epsilon_{32}=1.5\times10^{-4}\)	DERIVED below
the operator row sum \(\\|\mathrm{rowsum}(P)-1\\|_\infty\) (08.17's per-step guard) — a different quantity, not the conservation residual	float32	\(\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}\) at \(s=64\)	DERIVED below, in its own units; never \(\epsilon_{32}\)

field	shape / type	required	source
`tokens`	`int64[s]` (byte-level, vocab \(256\), or the bed's node tokens as multi-hot features `float32[s, d_in]`)	yes	task
`masks`	`bool[K+2, s]`, channel order (sink, goal, \(\mathcal A_1..\mathcal A_K\))	yes; \(K\ge2\)	task
`i_q`	`int64`	yes; \(i_q\in T\), every set \(<i_q\)	task
`moves`	list of \(m\) `("clamp", v_a, u_a)` with \(u_a\in T\), \(v_a\le i_q\) reachable, or `("rewrite", i, tokens_i')`	for \(\Delta z\), \(a^\dagger\)	task
`gamma_env`	`float32` in \([0,1)\)	for the discounted label only	task
`z_star`	`float32[s, d]` on \(P_{\rm env}\) at `gamma_env`	for \(\mathcal L_z\)	oracle on \(P_{\rm env}\)
`q_star`	`float32[K+2, s]` at \(\gamma=1\) on \(P_{\rm env}\)'s transient block	for \(\mathcal L_q\)	oracle (`bed_1.committor`'s form)
`dz_star`	`float32[m, s, d]` by the displacement identity on \(P_{\rm env}\)	for \(\mathcal L_\Delta\)	oracle
`a_star`, `y`	`int64`, `bool[m, K]`	for \(\mathcal L_a\), \(\mathcal L_c\)	oracle: \(a^\star=\arg\max_aq^{\star(0)}(a)\) on \(P_{\rm env}\)
`a_dagger_star`, `infeasible_star`	`int64`, `bool`	for \(\mathcal L_{a^\dagger}\)	oracle: the constrained argmin \(\arg\min_{a\in\mathcal F^\star}\max_{k\ge1}q^{\star(k)}(a)\) on \(P_{\rm env}\), \(\mathcal F^\star=\{a:q^{\star(0)}(a)\ge\max_{k\ge1}q^{\star(k)}(a)\}\); on \(\mathcal F^\star=\emptyset\) it is \(a^\star\) with `infeasible_star=True` and the term is skipped and counted (08.9, 08.16)
`seed`, `generator_id`, `sha256`	Ruling 7 triple	yes	task

kind	\(n\)	peak (B)	reserved (B)	peak/reserved (GiB)	`resident`
`softmax`	\(2{,}048\)	\(302{,}270{,}976\)	\(379{,}584{,}512\)	\(0.281/0.354\)	`true`
`softmax`	\(32{,}768\)	\(4{,}328{,}958{,}464\)	\(5{,}270{,}142{,}976\)	\(4.032/4.908\)	`true`
`arm_smprime`	\(2{,}048\)	\(874{,}290{,}688\)	\(1{,}061{,}158{,}912\)	\(0.814/0.988\)	`true`
`arm_smprime`	\(16{,}384\)	\(6{,}759{,}747{,}072\)	\(8{,}057{,}257{,}984\)	\(6.296/7.504\)	`true`
`arm_smprime`	\(32{,}768\)	\(13{,}484{,}888{,}576\)	\(14{,}979{,}956{,}736\)	\(12.559/13.951\)	`false`

verse	kill number (frozen)	price (GPU-s, dependency included)	requires	replacement verse / route
08.15	clause 1: pair counts under the rule the cell journals differ by \(>0.032\%\) (Ruling 3, R-15) — forfeits "at matched parameters" and every T1/T2 row; clause 2: \(>0.1\%\) (S-30) — the pair may not be journalled at all. Rule B: LM \(192/25{,}728{,}000=0.00074627\%\) matched, design \(3/4{,}769=0.062906\%\) UNMATCHED \(1.97\times\)	\(0\)	the two parameter counts; no draw, no arm	delete the gate heads and \(g_h\) from the module (Rule B and Rule L then coincide); kill is a different object: `torch.equal(O_with, O_without)` `False` on one draw at \(g=0\), \(0\) GPU-s. The MLP-widening replacement is deleted: at \(8\times1025=8{,}200\) parameters it is \(5.47\times\) the act Ruling 3 forbids by name
08.1	output tuple missing a field / shapes disagree with 08.1's table	\(0\)	one FOUND record of the arm's `kind`, plus the S-01 manifest field check	two-head form (H-q, H-z), kill stated per dtype
08.19	zero FOUND records with the arm's `kind` at end of Phase 2	\(0\)	the `results/` scan alone	08.1's two-head API
08.3	\(\rho(Q)=1\) with the sink declared, or conservation residual above the dtype's bar: \(\epsilon_{64}=10^{-8}\) float64, \(\epsilon_{32}=1.5\times10^{-4}\) float32; and, as a third and distinct quantity, the operator row sum above \(\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}\) at \(s=64\) — never \(\epsilon_{32}\), which carries the amplification $	T	\,\|(I-Q)^{-1}\|_\infty=1{,}500$ that a row sum does not, and which would admit a conservation residual of \(3.5715\times10^{-3}\), \(23.81\times\) its own float32 bar (DERIVED)	\(0\)
08.7	`route = neumann_K` on a committor cell; residual above 08.3's bar for the cell's dtype (\(10^{-8}\) / \(1.5\times10^{-4}\))	\(0\)	one declared draw	discounted read at \(\gamma_{\rm env}\)
08.18	leak probe \(R^2\ge0.5\) honest, or \(<0.99\) planted	\(0\)	the corpus and a CPU probe; no GPU	strip features → strike the task
08.9	clause 2 (census): R0 above the exact zero-information floor \(0.875\) by more than its Clopper–Pearson interval	\(0\)	the untrained-arm 512-draw census; no arm, no arena	committor-vector head only → goal rule only
08.10	class frequency of \(\phi\) outside \((0.05,0.95)\); \(\mathrm{sd}(\mu)\le0.05\); \(\hat\tau\) flat in \(t^\star\); 08.9's `infeasible_frac` or `s_discard_frac` above \(0.5\)	\(0\)	the 512-draw census at \(t^\star\in\{2,8,32\}\)	\(K=2\) isocommittor → hitting-time head
08.12	\(t^\star=8\) unplaceable at \(s=64\)	\(0\)	the bed as registered	reroute the dial → `bed_1` as cross-check only
08.14	clause 1 restated on the committor, not on placement: fraction of admitted draws with \(\max_{k\ge0}q^{(k)}_{i_q}>q^{({\rm sink})}_{i_q}\) below \(0.216\) (`[ASSUMED]`, DERIVED from the record's own sink-share interval \([0.362,1.000]\); the placement form at \(0.9\) is deleted — the geometry produces \(0.34416\) at \(p=3\) and \(0.57987\) at \(p=6\), so it fired on \(100\%\) of censuses); clause 2: \(\mathrm{sd}(q^{(k)}_{i_q})\le0.05\) on head \(0\) of the read layer	\(0\)	the 512-draw census	boundary rows in every layer with the head axis restored → learned membership (leak-probed)
08.22	\(\max\vert\Delta\vert>0\) or backward raises on `solve_triangular` over \(8\) repeats	\(\approx1\)	the certified device under both flag regimes	`warn_only` + noise floor → stored-\(P\) replay
08.2	`torch.equal` at the softmax corner `False`	\(\le1.7\)	one cell against the lane's own `softmaxAttn`	gate deleted, ChaCAL base (cited [V-eq], not [V])
08.4	solve vs dense \(>10^{-12}\) float64; `torch.equal(O(0),PV)` `False` float32	\(\le1.7\)	one draw at \(s=64\)	Neumann with certificate → dense per chunk
08.13	loaded-corner logits not `torch.equal` at the LM shape	\(\le1.7\)	a pretrained softmax checkpoint to load	corner frozen → parallel branch with V-24's three substitutions
08.11	one exceedance on \(1{,}024\) cells of \(\delta_{\rm bare}+\delta_{\rm mask}\) with \(\delta_{\rm mask}=\gamma\varepsilon/[(1-\gamma(1-\varepsilon))(1-\gamma)]\) (never the naive \(\varepsilon/(1-\gamma)\), which under-reports by \(4.74\times\)); \(\delta\Vert\tilde V\Vert_\infty\ge\) label sd	\(\approx2.4\) (\(0.04\) GPU-min)	\(1{,}024\) drawn cells at \(s=64\)	exact route only
08.8	column route \(\ge8\times\) one solve at \(m=8\), synchronize-bracketed, median of \(14\) after \(2\) warm-ups	seconds, band only — the median is unpriced and the timer is book 03's	N-02's synchronised timer	cached mixture → per-row control
08.6	\(r_{\rm softmax}/r_{\rm shape}\le2\) at marginal NRMSE within \(\mathrm{MDE}_8\)	\(\approx34\)	the \(\approx34\) s arena pair (shape + control, \(N=8\))	cost statement, kill at \(1.12\times\) the measured \(2.514\) ms \(z\)-channel total \(=2.816\) ms at \(n=2048\) (clock dispersion only; both model-derived candidates, \(2.474\) ms total and \(0.825\) ms increment, are printed in the verse and shown already crossed by the reference, so neither is frozen)
08.9	clause 1 (comparator): R0 flag rule \(\ge\mathrm{acc}_{\rm shape}-\mathrm{MDE}_8\) on \(6/8\)	\(\approx98\) — not \(0\): R0 is oracle-free, but \(\mathrm{acc}_{\rm shape}\) and \(\mathrm{MDE}_8\) are the arena's	the \(\approx98\) s eight-seed arena (08.16)	committor-vector head only → goal rule only
08.16	\(\hat\gamma\) PINNED and \(\mathcal L_q\) not moved by one seed sd	\(\approx98\)	the \(\approx98\) s eight-seed arena	committor-only (kill on the off-diagonal sup and the induced \(\varphi\)-NRMSE, never the full-matrix sup, which fires by construction at \(138.5\times\) the bar) → interventional pairs only
08.17	clause 1: `rowsum_residual` \(>\epsilon_{\rm rs}=s\cdot2^{-24}=3.815\times10^{-6}\) at any arena step, read beside `beta_final` — the inference is "the operator is not row-stochastic", and `beta_final` names whether the decay channel or the gradient channel moved it; the old inference ("the optimiser is not running the parameter groups") could not distinguish the defect from the Statement working as specified	\(\approx98\) — \(0\) GPU-s of instrument on top of the arena it reads	the \(\approx98\) s eight-seed arena	the frozen set is the Statement (\(\beta_h,\kappa_h,g_h\), the two gate heads, `requires_grad=False`) → T2 only → per-step cost law
08.5	\(\Lambda\le2.7055\) on \(6/8\) seeds, admissible only on seeds where \(\eta_h\) never left \([0,1]\) at either end, with `eta_excursions_low` and `eta_excursions_high` printed as two fields (the projection is \(\mathrm{clip}(\eta_h,0,1)\); a one-sided projection leaves the upper trap, which pins \(\gamma_{\max}\) at conditioning \(1024\) and a conservation residual \(25.6\times\epsilon_{32}\))	\(\approx115.4\) (\(\approx98\) arena \(+\le17.4\) ablation)	the \(\approx98\) s arena and the LR ablation cell	fixed \(\gamma_{\rm env}\) → annealed dial with committor supervision
08.20	K-9: exponent interval excludes \(2\) toward \(3\); ratio \(>3\) at \(s\le1024\)	\(206\)–\(537\) (band)	the synchronised, order-randomised \(s\)-sweep (S-66, book 03)	chunked solve → dense per chunk with \(n\) per \(s\)
08.21	memory law outside \([0.9,1.1]\); reserved \(>7.996\) GiB at the declared \(n\) (GiB against a card in GiB: `arm_smprime` at \(n=16{,}384\) reads \(7.504\) GiB reserved and `resident: true`; \(10.578\) GiB is the training-loop number and is not resident)	\(\approx720\) (\(\approx9\) GPU-min N-18 \(+\approx3\) GPU-min N-20)	the memory sweep over \(s\in[128,2048]\)	recompute backward → block checkpointing
08.17	clause 2: T1 ratio \(>1\) on \(6/8\) (interval excluding \(1\)), at the LM shape only, on BED-S at \(t^\star=8,K=2,m=8\), matched count \(25{,}728{,}192\) vs \(25{,}728{,}000\) (\(192\) parameters, \(0.00074627\%\), Rule B, no widened control), \(N=8\), skyline \(\lfloor\log_2t^\star\rfloor+2=5\) — unlicensed at \(d_{\rm model}=16\) under both counting rules, \(1.97\times\) and \(24.2\times\) the bar (08.15)	`NOT MEASURED` — needs the LM-shape step rate; the \(\approx26\) s and \(\approx64.8\) s figures are \(d_{\rm model}=16\) rates and are not carried (V-22)	the LM-shape step rate on the certified device (`scripts/k_cert.py`, book 05)	T2 only → per-step cost law (K-9, no optimiser in the chain)

08 — ARCHITECTURE: the module as an engineer builds it#

Preface — what this book decides, and the verse that decides it first#

Part I — The layer#

08.1 — The CEQ attention layer: interface and shapes#

08.2 — The causal row-stochastic three-corner operator \(P=W_{\beta,qk,g}\)#

08.3 — Boundary rows: the goal set, \(K\) constraint sets and the declared sink as identity rows#

08.4 — One lower-triangular solve with \(K+2\) value channels, and the read#

08.5 — The horizon dial \(\gamma\): parameterisation, initialisation and the interpolation from softmax to the committor read#

Part II — The reads#

08.6 — The next transient state \(z\) and the mixture read \(O\)#

08.7 — The committor vector \(q^{(\bullet)}\): the \(K+2\) channels through the exact solve at \(\gamma=1\) on the transient block#

08.8 — The displacement \(\Delta z\) under \(\mathrm{do}(a)\): the interventional re-solve and the Sherman–Morrison column route#

08.9 — The safest move: the argmin over candidate moves of the maximum constraint committor, goal first#

08.10 — The equilibrium-phase head: which absorbing set the mass flows to, the decision point, and leaving the transient class#

08.11 — The Neumann certificate as an exposed output#

Part III — Transient-state traversal#

08.12 — A sequence read as a trajectory through transient states toward an absorbing equilibrium#

Part IV — Integration into a transformer block#

08.13 — The block: pre-norm residual, MLP, multi-head, and the \(\gamma=0\) corner that loads a pretrained softmax stack unchanged#

08.14 — Which layers carry boundary rows, and how \(K\) sets are declared per task#

08.15 — Matched parameters (Ruling 3): the count, and what the CEQ block adds#

Part V — Training#

08.16 — The training objectives and their weights#

08.17 — The optimiser regime and the record's numbers; T1/T2 as the two measured quantities#

Part VI — The data interface#

08.18 — What a task supplies: tokens, sets, interventions, labels#

Part VII — The inference API#

08.19 — One forward call, and what it returns#

Part VIII — Budget#

08.20 — The per-layer, per-step FLOP and byte model on the certified RTX 4060 Laptop and on a Kaggle T4/P100#

08.21 — Memory residency of the shape at the softmax corner, and the \(n\) per \(s\) rule#

Part IX — Determinism and reproducibility#

08.22 — The identity manifest, the seed rule, and the bitwise forward regime#

Closing tables#

Census rows closed#

Kills, cheapest first#

Attacks answered#

A-1 · verse 08.2 · severity strike · mechanism V-25, V-3, P-7 · replacement_survives: true#

A-2 · verses 08.13 / 08.15 / 08.17 · severity strike · mechanism M-8, V-25, D-3 · replacement_survives: false#

A-3 · verse 08.15 · severity strike · mechanism Ruling 3, M-8, D-1, R-SKY · replacement_survives: false#

A-4 · verse 08.3 · severity strike · mechanism V-17 (a threshold imported out of its units), M-2 · replacement_survives: false#

A-5 · verse 08.1 · severity strike · mechanism V-11 inverted (a kill that fires at every real draw), V-17 · replacement_survives: false#

A-6 · verses 08.3 / 08.4 · severity strike · mechanism V-3, V-23, B26 · replacement_survives: false#

A-7 · verse 08.1 · severity strike · mechanism B26 (the gap the book exists to close), P-4 · replacement_survives: false#

A-8 · verse 08.9 · severity strike · mechanism V-16 (three outcomes, never a silent number), V-12 · replacement_survives: true#

A-9 · verse 08.9 · severity strike · mechanism P-4 (a link that does not exist), B26 · replacement_survives: true#

A-10 · verse 08.9 · severity strike · mechanism V-25 (a theorem whose hypothesis no draw satisfies), L-FLOOR · replacement_survives: true#

A-11 · verses 08.9 / 08.16 / 08.19 · severity strike · mechanism D-2, M-1, V-8 · replacement_survives: true#

A-12 · verse 08.11 · severity strike · mechanism L-CERT, V-3, V-17 · replacement_survives: true#

A-13 · verse 08.12 · severity strike · mechanism D-2, V-25, V-11 · replacement_survives: false#

A-14 · verse 08.14 · severity strike · mechanism V-10 (a gate satisfied by construction), V-11 · replacement_survives: true#

A-15 · verse 08.19 · severity strike · mechanism V-16, V-10 · replacement_survives: true#

A-16 · verse 08.21 · severity strike · mechanism V-17, V-22, P-8 · replacement_survives: true#

Consequential edits this batch made outside the verses the findings name#

Open after this batch#

B-1 · verse 08.17 · severity strike · mechanism M-8, P-8, V-17 · replacement_survives: true#

B-2 · verse 08.5 · severity strike · mechanism V-9, V-10, B12 · replacement_survives: true#

B-3 · verse 08.16 · severity strike · mechanism P-1 (a number with no class), V-16, B26 · replacement_survives: true#

B-4 · verse Preface (items 9, 15) · severity strike · mechanism P-4 (claimed scaffolding that does not exist), B26 · replacement_survives: true#

B-5 · verses 08.7 / 08.10 · severity repair · mechanism L-EQ, P-10, P-11 · replacement_survives: true#

B-6 · verse 08.1 · severity repair · mechanism B26, P-4 · replacement_survives: true#

B-7 · verses 08.15 / 06_PREDICTIONS · severity repair · mechanism M-2, Ruling 3 · replacement_survives: true#

B-8 · verse 08.21 · severity repair · mechanism M-8, P-8 · replacement_survives: true#

B-9 · verses 08.20 / 08.21 · severity repair · mechanism M-8, V-22, B15 · replacement_survives: true#

B-10 · verse 08.9 · severity repair · mechanism M-8, P-8, D-3 · replacement_survives: true#

B-11 · verse Closing table "Kills, cheapest first" · severity repair · mechanism CHARTER §2 depth rule, D-3 · replacement_survives: true#

B-12 · verse 08.9 / Limits · severity repair · mechanism P-1, B10 · replacement_survives: true#

B-13 · verses 08.1 / 08.11 · severity repair · mechanism L-CERT, D-3, P-4 · replacement_survives: true#

B-14 · verses 08.13 / 08.22 · severity repair · mechanism B13, Ruling 1, V-16 · replacement_survives: true#

B-15 · verse 08.6 · severity repair · mechanism P-1, M-2 · replacement_survives: true#

B-16 · verse Preface (item 6, line 12) · severity repair · mechanism P-1, P-7 · replacement_survives: true#

Open after this batch#

A-1 · verse 08.2 · severity `strike` · mechanism V-25, V-3, P-7 · `replacement_survives: true`#

A-2 · verses 08.13 / 08.15 / 08.17 · severity `strike` · mechanism M-8, V-25, D-3 · `replacement_survives: false`#

A-3 · verse 08.15 · severity `strike` · mechanism Ruling 3, M-8, D-1, R-SKY · `replacement_survives: false`#

A-4 · verse 08.3 · severity `strike` · mechanism V-17 (a threshold imported out of its units), M-2 · `replacement_survives: false`#

A-5 · verse 08.1 · severity `strike` · mechanism V-11 inverted (a kill that fires at every real draw), V-17 · `replacement_survives: false`#

A-6 · verses 08.3 / 08.4 · severity `strike` · mechanism V-3, V-23, B26 · `replacement_survives: false`#

A-7 · verse 08.1 · severity `strike` · mechanism B26 (the gap the book exists to close), P-4 · `replacement_survives: false`#

A-8 · verse 08.9 · severity `strike` · mechanism V-16 (three outcomes, never a silent number), V-12 · `replacement_survives: true`#

A-9 · verse 08.9 · severity `strike` · mechanism P-4 (a link that does not exist), B26 · `replacement_survives: true`#

A-10 · verse 08.9 · severity `strike` · mechanism V-25 (a theorem whose hypothesis no draw satisfies), L-FLOOR · `replacement_survives: true`#

A-11 · verses 08.9 / 08.16 / 08.19 · severity `strike` · mechanism D-2, M-1, V-8 · `replacement_survives: true`#

A-12 · verse 08.11 · severity `strike` · mechanism L-CERT, V-3, V-17 · `replacement_survives: true`#

A-13 · verse 08.12 · severity `strike` · mechanism D-2, V-25, V-11 · `replacement_survives: false`#

A-14 · verse 08.14 · severity `strike` · mechanism V-10 (a gate satisfied by construction), V-11 · `replacement_survives: true`#

A-15 · verse 08.19 · severity `strike` · mechanism V-16, V-10 · `replacement_survives: true`#

A-16 · verse 08.21 · severity `strike` · mechanism V-17, V-22, P-8 · `replacement_survives: true`#

B-1 · verse 08.17 · severity `strike` · mechanism M-8, P-8, V-17 · `replacement_survives: true`#

B-2 · verse 08.5 · severity `strike` · mechanism V-9, V-10, B12 · `replacement_survives: true`#

B-3 · verse 08.16 · severity `strike` · mechanism P-1 (a number with no class), V-16, B26 · `replacement_survives: true`#

B-4 · verse Preface (items 9, 15) · severity `strike` · mechanism P-4 (claimed scaffolding that does not exist), B26 · `replacement_survives: true`#

B-5 · verses 08.7 / 08.10 · severity `repair` · mechanism L-EQ, P-10, P-11 · `replacement_survives: true`#

B-6 · verse 08.1 · severity `repair` · mechanism B26, P-4 · `replacement_survives: true`#

B-7 · verses 08.15 / 06_PREDICTIONS · severity `repair` · mechanism M-2, Ruling 3 · `replacement_survives: true`#

B-8 · verse 08.21 · severity `repair` · mechanism M-8, P-8 · `replacement_survives: true`#

B-9 · verses 08.20 / 08.21 · severity `repair` · mechanism M-8, V-22, B15 · `replacement_survives: true`#

B-10 · verse 08.9 · severity `repair` · mechanism M-8, P-8, D-3 · `replacement_survives: true`#

B-11 · verse Closing table "Kills, cheapest first" · severity `repair` · mechanism CHARTER §2 depth rule, D-3 · `replacement_survives: true`#

B-12 · verse 08.9 / Limits · severity `repair` · mechanism P-1, B10 · `replacement_survives: true`#

B-13 · verses 08.1 / 08.11 · severity `repair` · mechanism L-CERT, D-3, P-4 · `replacement_survives: true`#

B-14 · verses 08.13 / 08.22 · severity `repair` · mechanism B13, Ruling 1, V-16 · `replacement_survives: true`#

B-15 · verse 08.6 · severity `repair` · mechanism P-1, M-2 · `replacement_survives: true`#

B-16 · verse Preface (item 6, line 12) · severity `repair` · mechanism P-1, P-7 · `replacement_survives: true`#